I just don't know what my next step should be. Am I doing the right thing by moving out? I do see ways in which I neglected our relationship. I know I can do better if given another chance. But I'm afraid that trying to ask him for another chance would just ruin my chances.
Don't ask for another chance, it implies "I want another chance at failing", subconsciously at any rate. Agree with this decision to separate even if it's not what you want.
Move out if you want to move out, stay if you want to stay, it is just a place that you live, nothing more. If you see the ways in which you neglected the relationship, improve on those things without requiring to be in a relationship with you. Don't pressure him, let go of your expectations on how he should act, raise the bar for yourself and do the things you think would make him feel better without coming off as needy & insecure, do it from a loving place and again when you do those things, don't expect a certain response or reaction. It took him alot of time to get to this place where he is right now (you too) and allow him the time to come back to a new place with you, somewhere he is comfortable, somewhere you are comfortable - otherwise you doom yourself to the same failure.