Yes you are correct. The grammar for this is known as a relative clause. 'Which' in this example is a 'relative pronoun', used to make one longer sentence from two shorter sentences: I had to accept it as my room or move out. I was not financially stable to move out at the time. You could also say "I had to move out, which I was not financially stable to do, or accept it as my room." The position of 'which' is always after a comma, and always after the thing you are adding information about. Notice where the commas go in each example though!

Yes, the 'which' refers to 'moving out'. The problem with this sentence is that there's a word missing. It should be: I either had to accept it as my room or move out, which I was not financially stable enough to do at the time.