very sorry for the length

this is happening very fast.
one more question
is there anything i can do about her “fiancee” living with her and my daughter if i cant get custody. like i said before they’ve been seeing each other for only a month and not only has he been introduced to the baby, but i beleive that he wants to adopt my parental rights.
i have NO idea whats going on.

in august my now ex fiancee and i moved to NC from ny, i lived with her, my daughter and her parents for about two months, i moved back to ny because we broke up and i had nowhere else to go (i had trouble finding work, eventually found it, but it wouldnt pay the bills) i lived with my daughter for her enitre life before that day, and i considered myself as her primary caregiver because i was the only one with an income outside of state assistance, which she had.

now its april, my daughter will be 11 months old (born in may) and my ex wants to move again to florida, i already pay support in NC, and i will continue to do so until im told to stop. now my ex has met another man, and after seeing him for a month decides that they are gonna get married, i dont know this guy and i fear that she dosent either, i have no idea what he does or how much money he has, or even if he knows what he wants, i havent yet spoken to him, or her, for that matter since she told me that my calls were unwelcome and that i was “dead” to her and my daughter.

she has started seeing an attorney to setup visitation priveleges and has come up with some ridiculous things to say about me
i recieved a summons claiming that im a drug abuser, that my freinds are drug abusers, that i cant provide a safe home for her (we both have the exact same situation, i live with my parents right now to get back on my feet after placing all my income in the two moves that i had to make)and also that she was my daughters primary for her entire life.

she wants visitation every other weekend for half days, supervised.
this is ludicrous in my eyes, and i want full custody for her because i have better means to raise her financially (she does not have a job right now and lives off of whatever her parents and boyfreind give her, also my child support), as well as an extended family to help me care for her (the only family she speaks to are her parents, older folks, retired mother suffers from emphysema, and her brother, who is twenty years older and lives two counties away, i can provide my parents, both professionals and in good health, and my grandparents who are retired and also in good health, and two uncles and two aunts, all with kids already) , not to mention a better standard of living, (they live in a very low income area of NC in a trailer {modular home} and i live in an upper middle class neighborhood in NY in a two story house), better schools and lower crime rates also.

shes also taken to slandering me and defaming my character online, but i can handle that.
my questions are:
is she allowed to move even further away from me?
can i restrict her movement to inside of nc or in her home county?

i dont want my daughter to be a ping pong ball and i definitely dont like the fact that my ex and her new man are moving so fast, as a result this will be the third time in five years shes gotten engaged and this trend will likely mess my child up.

shes not a bad mother at all and i dont have anything against her getting a boyfreind, im just frustrated at the speed in which this is all going. i dont want to ‘take the baby away’ per se, i just believe i can offer her a better life up here in the city with better support given the fact that my we will both have to work to earn money and her parents are in deteriorating health.

do i have a shot at all at custody?

all i wanted was to be amicable, but she apparently doesn’t want peace here.