Highthoughts

Your situation, and the advice you need, is accurately given by Elder Matthew Carter during the Sunday afternoon session of the October 2018 General Conference in a talk entitled Wilt thou be made whole.
https://www.churchofjesuschrist.org/study/general-conference/2018/10/wilt-thou-be-made-whole?lang=eng
As I have counseled with others seeking to repent, I have marveled that people who were living in sin had difficulty making correct decisions. The Holy Ghost would leave them, and they often struggled to make choices that would bring them closer to God. They would wrestle for months or even years, embarrassed or frightened of the consequences of their sins. Often they felt that they could never change or be forgiven. I have often heard them share their fear that if their loved ones knew what they had done, they would stop loving them or leave them. When they followed this line of thinking, they resolved to just keep quiet and delay their repentance. They incorrectly felt that it was better not to repent now so that they would not further hurt those they loved. In their minds it was better to suffer after this life than go through the repentance process now. Brothers and sisters, it is never a good idea to procrastinate your repentance. The adversary often uses fear to prevent us from acting immediately upon our faith in Jesus Christ.
When loved ones are confronted with the truth about sinful behavior, while they may feel deeply wounded, they often want to help the sincerely repentant sinner to change and to reconcile with God. Indeed, spiritual healing accelerates when the sinner confesses and is surrounded by those who love them and help them to forsake their sins. Please remember that Jesus Christ is mighty in how He also heals the innocent victims of sin who turn to Him.13
With your husband being a leader in the church I expect that he would have a sound understanding of the doctrines of repentance and forgiveness and act accordingly. Also, from my own experience, I know that our perspective on repentance looks very different before we repent than after we repent. Before we repent, we often see it as a scary, embarrasing, fearful, painful process. After we have repented, we often see repentance as a great miracle for which we are truly grateful. Satan doesn't want us to repent so he really focusses on the fear and embarassment.  
 

Hi,  I can't speak for your husband.  I can only tell you of what I went through.
My wife and I didn't have the best of marriages for a long time.  She ended up have a year and a half long emotional affair with a colleague.  I didn't find out until later.  
I was emotionally devastated by this.  I could barely function for 4 months.  I very much contemplated divorce.  But our Heavenly Father asked me to take a different path.  It was by far the hardest thing that I have ever experienced.
I didn't stay married for the kids.  I stayed married to love my wife.
To me staying married for the kids is just lying to the world.  It is just compounding the sin.
It took me the better part of a year to learn to love her again.
I certainly can't tell you how he will react.  There is a chance that he will want a divorce.  But to set things right, you have to set things right with the person her on Earth you should love the most and that is your spouse.
As much as it hurt, I am grateful that she was honest with me.  Our marriage would probably continue to suck had I not found out.  Is our marriage great now?  No, but it is better today than it has been in a long time and getting better every day.
One thing she did for me that helped me get over it was that she let me read the whole thing.  She answered all of my questions.
If you want this to work for you, I suggest that you in no way shape or form put any blame on him directly or indirectly on him.
It has almost been two years.  It sucked, but I feel healed and I love her.

I'm trying to catch up on reading the Ensign.  Last night I read an article from October's issue, linked below.  I know it's not exactly addressing your situation, but I think a lot of the concepts and questions could still be helpful to you.
https://www.churchofjesuschrist.org/study/ensign/2019/10/dating-and-pornography?lang=eng

You have great advice above.
I'd like to add this:
Trust your husband.  Trust that he cares about your children as much as you do.  Don't dishonor him by ripping him of his agency to do what is best for the eternal family when he has not actively done a thing to destroy it yet.

No it's not. God does not want you to go to hell, and you won't be doing yourself or your children any favors by not completing the repentance process. Remember what God tells us in John 3:17
"For God sent not his Son into the world to condemn the world; but that the world through him might be saved."
Repentance can be tough, but God sent his Son Jesus Christ into the world to save you as an individual, as well as all of us. His whole goal is to bring you eternal life, not have you suffer in misery for the rest of your mortal life. The only person who is telling you that you deserve eternal misery is the devil. Tell him to go back to hell where he belongs, take a deep breath, and find some time to tell your husband. It will be difficult and he may well be very hurt. But you cannot build a solid relationship on lies. I would rather my wife tell me something like this so we could work it out and come out stronger on the other side. Especially since it did not get physical, I think if your husband is truly the great guy you think he is he will be able to forgive you in time. Talk to your bishop too. You will be amazed at the feeling of relief you receive the moment you talk about this to an authorized servant of God. God is waiting to forgive you sister. The question is will you let him? I know you can do this and I will be praying for you. 

No it isn't. Heavenly Father's goal is to have ALL His children return home - including you.
Once we return home and have a full understanding of our Eternal nature and the Savior's love, the suffering your children (may) experience in this life will pale in comparison to the sadness they will feel if you can not join them in the Celestial kingdom. Remember the joy over one soul that repents? You are that one soul at this time. Any price that may be paid today on earth (marriage, husband, children) will be swallowed up in joy and forgiveness in heaven.
So, do what you need to do with humility and a broken heart. Tell your husband you need to tell him a story, and he will have to tell you how it ends. As a couple, you may or may not choose to share this with your children. But whatever happens, make and stick to a commitment to confess to your Bishop and repent while you patiently and meekly endure all things. It's ALWAYS worth it.

This is a good thing to lay bare to your bishop.  Go seek his counsel.  You need to do more than just figure out how to handle this in your marriage, you need to avail yourself of the atoning sacrifice of Jesus Christ and be cleansed of this sinful burden.

I have had an emotional affair. Looking back I believe the reason it started was in dealing with depression. I'm not a very good communicator with my husband and often feel like he is annoyed or upset with me. I had a friend from high school that I felt safe talking to about absolutely anything, partly because there was no investment. I could say anything to him and if he didn't like it, oh well. He wasn't my partner so there was no risk. But it developed into more as I went to him to get the attention/comfort/ etc. I should have sought from my husband. It went too far and we had a "sexting" affair. We never had sex, but there were photos/videos exchanged and we met in person as well.
My problem is this. My husband is an amazing man. He is a leader in the church and a wonderful father. Our issues are all about communication and my inadequacies in everything. We have three young children. I fear upon learning of my affair that he will immediately want a divorce because that's the one reason Christ said it is okay to divorce. I fear he will see it as just the consequence of my action. And I fear my children will suffer greatly.
I'm considering just carrying on as if it didn't happen. Repenting as much as I can and just trying to be the perfect person on the outside that I failed to be on the inside. Isn't it better to go to hell myself than for my kids to suffer through losing a parent and all of that?
Any advice is appreciated. Anyone out there survive something like this?