Wow, that is really tough. I agree most with Christy, that telling them honestly what is going to happen is best. Also, what kind of cancer do you have and what is your prognosis? Not that you need to tell us, but this can inform what you do. My mom is a 10-year survivor of breast cancer and had a small lung tumor taken out this past summer, so depending on your cancer you could do just great.
When I think of what I would have to do for my 2-year-old and upcoming newborn if I or my husband were dying, dead or severely incapacitated, here is how I think of it: I would want to keep as much continuity in their lives as I could, because losing a parent is DEVASTATING to a child. Seriously, it is the worst thing that could happen. Of course my morbid fantasies usually center on a sudden car crash, not a prolonged illness, which gives you time to plan and decide how to deal with it.
For your cancer plan, I would recommend getting a friend to go with you to your doctor's appointments and go with lists of questions. You will be so overwhelmed with information and perhaps emotion that you may not be able to quiz the doctor like you need to and in the car ride home you'll be wondering what the heck she or he said. So get a friend to go and ask the hard questions for you if she doesn't think they were answered. Go with pen and paper and write down the answers. If you are going to have surgery and recovery, so essentially short-term incapacitation, see if a friend or family member can come down for a few weeks so that your kids can keep going to school and they can maintain some of their normal routine. If you are going to have to have long-term incapacitating treatment, consider moving closer to family or whoever so that the kids can have the support you can't give them until you are better. Also, if your prognosis is poor, this will enable them to build a new routine and new emotional support if you die.
If surgery is the typical option, I would recommend being aggressive and taking the best option to reduce further occurrence. Since you have two kids, I don't think playing it safe with a minor surgery is the best option, but that's your decision and one to make on the advice of your doctor.
Get a second opinion and make sure your doctor listens to you. Again, go to your appointments with a friend, pen, and paper, and write everything down. My mom did this and it REALLY helped. It will also really help you tell your kids what is going on.
Also, find a confidant. Either a friend you trust, a pastor, a counselor, someone you can talk to about your fears. It's ok for your kids to know you are scared, it's ok for them to know what is happening, but it's best if they don't see you freak out. You can't be strong all the time, so find someone you can cry with or in front of. Now is the time to ask for help. People LOVE to help, they really do. It makes them feel honored and needed, and we all love that. Ask your friends for rides to the doctor's, help with some meals, errand-running, things like that. If you spread out the help, people love a mission, and it will bolster your energy to realize you have such a strong support group.
Good luck with everything! If you need to email, go ahead.