ok, so before I could give you advice on how to remedy your situation, it would be useful to know a few things...
1) how old is your dog now?
2) how old was your dog when you got him/her?
3) how long did your dog live with the two of you before the breakup/change of circumstance?
4) do you work from home/what do you do when you have to be out of the house, and how does your dog react to that?
so, one of the hardest things, especially for first time dog owners, is teaching your dog to be alone. but it's very important. if your neighbours are an issue (we had a puppy in a shared building, so trust me, I know how stressful it can be) the best thing to do is to speak to them. let them know times when you're going to be working on separation and that during those times your dog will be howling. if possible, even arrange those times around their schedule so they're not at home while you're doing it. and then it's a matter of building up the amount of time your dog can be alone. crate training can be very useful, but once you start it's important to persevere with it. at first, your dog might not like it. ours didn't. but now, seven years later, we still have the crate (we never close the gate) because she uses it as intended - it's her space where she can sleep or just chill out. so the rewards in the short term - being able to leave your dog alone; easier housetraining - plus the long term benefits make it a win-win. but as I say, you have to stick with it through the initial period of resistance that most dogs go through.
that being said, everyone's situation is different. however, dogs that develop severe separation anxiety are not happy dogs. they're not even really happy when they're with you because the anxiety of being apart never really goes away. and an anxious dog can be a destructive dog when alone and an aggressive dog when out in the world. so it's an issue that really needs to be dealt with early if at all possible.