Can I get full custody of my son?

My sons father is not involved in his life, he might see him once a month iff that. he gave me 80 dollars the beginning of may but he doesnt usually help out he doesnt buy him diapers or anything. I lost my job and he didnt even offer to help out. My son calls another guy dad because his dad is not around, My son does this on his own I did not force him to call him dad. my son is 14 months and he is afraid of is real father he crys and he does not know him. I try to get him to be involved but hes always to busy. His own mother said I should take him for child support. Could I get full custody. If something happens to me his father is not stable enough to take care of him what do you think? My son is with me all the time his father never asks about him