20+ years.
Sorry. 16 years.
Wow... That's up there. I recently replaced mine that was also about 16 years old. It's time to replace yours. Leave everything on/functioning and put a pan, bucket or towel underneath it to catch the dripping water. Check it every several hours to make sure that the leak isn't any worse and confirm that you're able to catch the water from doing any other damage to your house. Nothing is going to bust unless you start hammering on it or try loosening/tightening pipe fittings or valves. You should be fine for days/weeks/months depending on the extent of the leak.
If the pilot goes out, just relight it to keep the water hot for showers and "stuff". If the water starts to affect the pilot, there's a check valve that should automatically shut the gas off when the pilot goes out, so you should be fine. Don't worry too much about water dripping on the burner, but I wouldn't let that go too long... as in months. The water could cause the burner to rust and result in a little carbon monoxide. You don't want to be responsible for global warming.
What is American Homeshield? You might do better just calling Lowes or Home Depot. Better yet, go shopping for a new water heater this weekend and see if you can schedule installation next week. I think I paid about $1,000 last year to replace a 40 gallon. That included installation along with removal/disposal of the old one. I did get a higher end model, but you should plan on spending at least $800 for a 40 gallon.
Hope this helps...