Worth trying for custody or shared custody?

My daughter is 4 and resides in OH. Her Mother and Stepfather have moved a total of 13 times since her birth and that's not even counting the residences I don't know about. They've been evicted numerous times and it's all public record, some evictions filed with complaints. I also have been able to find out that the stepfather has been in jail before (not for sure what for) and that he works for cash, has no solid income. The mother doesn't work either. They collect state assistance and are all on Medicaid. I on the other hand, live in another state and have been in same career 15 yrs. I own my own home and have stable income and can provide for her a better life more than these 2 losers. My question is - is it possible for me to get full custody or even worth a shot? Maybe shared custody, where I get her all of school year? The mother also has a son who resides FT with his bio father.except on some wkends, she gets him.That says alot. Daughter needs stability!!