Don't use cell phone batteries . PERIOD. It's a long story but the short version is they are made for VERY LOW POWER CIRCUITRY.
Most of the stuff you described does not begin to fall into that category and the L293 IS JUST PLAIN CRAZY to try to run from a cell phone battery. It's so crazy I won't even talk about it . Just forget it .PERIOD. Get some RC batteries , or just one. Get a 2S 2200mAhr
battery, 25C to 40C. Make sure to use a Lipo charger. If you don't want to use Lipo's then use NiCad. They're heavier but don't have
some of the hazards that Lipos have. Ask around and you'll find out what I am talking about. Just to put it into perspective. If for some reason you weren't paying attention and accidently shorted the leads of a Lipo battery, it would burst into flames instantly,
just like that , every time. I acidently punctured one with my pocket knife trying to separate two of the cells and it burst into flames right in my hand. Yes, I dropped it like a hot potato. and it scorched the parking lot asphalt. That's Lipo's for you.