First off, I really doubt the home warranty will cover it. It only covers damage from something sudden....like interiour damage due to a water break. What you have is normal wear and tear....things wear out....most home warranties don't cover that.
Regarding your light problem....sounds like you don't have a volt meter...time to go buy one....it will tupically pay for it's self the first time you troubleshoot a problem.....especially if you are the shotgun type....(replace stuff until it works).
Get a meter....with the switch on, see if your getting voltage to the ballasts....if so, and you know the bulbs are good....time for a new ballasts....
No voltage? Go back to the switch...if nothing there...keep going back until you find where it starts....
On last thing regarding home warranties.....you really need to avoid using them if possible....they keep a data base of that stuff.....if you use it, your in the data base....so the next time you go to get an insurance quote......your claim (whether it was paid or not) shows up....so if you filed a claim for water damage due to a leaking pipe.....some insurance carriers may not want to take you on....why? Because you now may be a mold risk.....in other words....a potential future claim to them....