I have an Atwood water heater# GC10A-3E. I have

Sounds like some adjustments are needed. Air shutter for one, and ignitor/electrode. Have you by chance observed outside with cover open when it tries to re-light? My guess is that it is sparking/trying to ignite, but adjustments are needed for proper ignition. A good cleaning of burner tube, electrode, and proper adjustment of electrode & air mixture should cure your problem.

Copy following link for Atwood water heater resources:
http://waterheatertimer.org/Atwood-water-heater-resources.html
http://waterheatertimer.org/How-to-test-water-heater-element.html
Add a comment and say what you discover

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My guess is that you may have or have had a spider web or other debris in burner tube that caused flame to back up, thus burning some wiring but, have someone experienced check it and clean it. The wiring if not burned too bad may be able to be repaired/spliced.

Sounds like the new thermostat and ECO you installed, You may have got them mixed up. The ECO will have a small reset push button in between the 2 spade connectors. The ECO should be second in line, as in power to it comes from the thermostat. Hot (black) line from switch goes to thermostat first, then from opposite spade on thermostat to spade on ECO, then out from ECO to element. Double check wiring or positioning of thermostat and ECO, and correct as needed, and make sure to push reset in center of ECO, then try again. Be sure both are touching up against tank well too.
The thermostat has a little lower temperature setting, and if it fails, ECO takes over and cuts out at a little higher temperature. That's why I suspect they got mixed up, otherwise thermostat (without reset) should have cut power at proper temperature, unless it's faulty, or not touching against tank properly.

Let me know how you make out and what you found.

Coming out of the back should be a 1/2" NPT short nipple, or a check valve with 1/2" NPT threads. ( NPT = National pipe thread. Just ask for a 1/2" galvanized pipe nipple the right length.(short, or 2" usually) If it's a check valve, it will usually be brass or plastic/nylon. I prefer galvanized just because the tend to hold up better. Most water lines connect to that, unless yours has a different fitting such as a female flare going to a male flare fitting in tank. Either way, you'll have to get old one out and match it, to allow for water line fitting to re-attach, or change end of water line to proper fitting to attach to nipple. Without seeing exactly what you have, I can't explain exactly what you need to do.

suggestion, ck your electric element to see if worn out.
you can ck with an ohms tester for current.

Obviously breakers are marked wrong, and good thing you checked. I always double check as well, seeing how I've found many mislabeled right from factory, or perhaps previous owners switched them and never changed original labeling. I need to ask though, are you sure the light is not just for the electronic ignition for the propane? Most have a red light to indicate propane side did not light when tried, and no light for hydro side. Have you also checked for another switch on side of cover at back of water heater where you see hydro cable going in to it? I guess I should ask as well, if you've owned this motorhome for a while, or is it new to you? I'm not trying to be sarcastic, but sometimes I forget to ask that and then find out they're new at this.