In time the guilt will lessen but the next time you are tempted to cheat, remember how you feel right now. That should stop you going forward. Everyone should have another chance but you have to actually take care to not ruin it.
As for getting caught, would there be a reason to get caught? The act of cheating itself isnt the worst part of the situation, it is the other person finding out and the hurt they feel. Depending on the situation or circumstances of the cheating, I say keep your mouth shut. No sense hurting someone because of your guilt. You may feel relieved but they will feel terrible. It's not a good thing to do to the person you love.
It's understandable that you feel guilty, especially if your partner has been faithful to you so far. I believe the only way to get rid of that feeling of guilt is by being 100% honest and exposing your mistake to your partner. This, as you may know, will hold a number of consequences, but chances are your partner will be more likely to forgive you. It will be a lot worse if they ever find out some other way. I hope my advice was useful. Best wishes :)
I don’t really have an answer for you, but I wanted to say I know how you feel as I am in the same boat. I found this website by searching for an outlet for my feelings on this same situation. I started a sexual relationship with a coworker about 2 months ago. We are both married to other people. Neither of us has told a soul, but the emotional impact and guilt is weighing heavily on me. You posted this a while ago, so hopefully you have been able to sort out your feelings. Take care friend.