R.J.
I doubt she was ever homesick. At this young, home is where your mom and your blankie are.
What I WOULD suspect is that when you left you were sad and angry and hurt and confused and so she picked that up off of you. Small children are like mirrors of what's going on with us... because they pattern their reactions off of ours. (Ever see a kid fall, look to their mom before the react and then either wave and brush themselves off, or burst into tears? When they're hurt they just react, when they've gotten scared they look to us for how they should react. If it's no big deal to us, no biggie to them. If we've leapt half out of our chairs and have an OMG look on our face they freak out.)
At this age, will she understand why and what you're doing? No way on earth. She's not going to have a clue.
Your daughter may be naturally shy, or it may be learned shyness out of fear & insecurity because of the drama/danger/fear/anger at home. Parents fighting is TERRIFYING to small children. Heck, even as an adult if your parents have a massive argument in front of you it can be scary. But as a child you are their everything.
Either choice you make will be hard on both of you. If you leave, it will be scary for both of you (you for ALL the reasons in you mind and heart, and her plain and simple because you're scared). If you stay... well... be prepared to not only deal with ONE quasi-abusive alcoholic, but also a child who learns to react JUST like daddy. Ever notice how much an alcoholic's temper tantrums are like a toddlers, only much much worse? She'll pattern after him, and her temper tantrums will have ALL of the elements his does... right down to how she views YOU. It will be 10x the work of the normal terrible 2's or terrible 3's... because she'll be having the *best* example to follow and learn from. Wheeeeee. SO much fun.
I could go on for awhile, as I have a temper tantrum throwing alcoholic husband. Sober for a year, and his temper is worse than ever. LOVED it btw this spring when my son spent 2 hours sobbing his heart out in the garden because while playing CATCH with daddy, he threw too low, hit his father in the leg with the ball and so "daddy" exploded. Yelling, whining, throwing his mit and ball into the bbq so hard it knocked the lid off. Which infuriated my husband even more... so he kicked the entire BBQ over. Then stomped off into the house swearing. I almost choked him. I DID kick him out. ((BTW... my husband is clueless. He describes the incident as "getting a little frustrated, and spoke sharply". A**hole)). Ugh. I can't even think about it without almost breaking a tooth from clenching my jaw so hard.
You have no idea how badly I wish I had left him for good the first time I left when our son was 3. No idea. Over and over I weep at the phrase that "It's better to come from a broken home than to live in one." No freakin lie.
Edited to Add: Childsupport and Custody are 2 totally separate things. A parent will *always* have to pay support, even if the courts say no contact, unless it's 50/50 custody. You're not paying to see your child, you are paying for your share of the cost of raising your child.