You will accrue illegal presence time from the day you quit your L1B job until your new employer places the application in for the H1B.
If things go smoothly (and you don't try to leave and reenter the US in the interim), you should be OK. Understand, that H1B is not the smooth sailing your L1B visa was. There are caps and wage requirements for the H1B that don't exist on the L1B.
You should sit down with an employment law attorney about the non-compete agreement. We can't commenton the implications of that or if it is even enforceable.

