Welcome BadChoices,
2 months with someone is a very short time, and I agree with the advice of the other posters. My neice who is currently living with her bf of one year has started to have issue with his drinking, and even after one year I am telling her that it may not be a pretty road and she is still young and she may want to move on.
My AH, who is currently struggling with a relapse, used to not drink when there was a constraint on him, like work the next day, or in your case, running out of money. Then even those boundaries get pushed, like missing a day here or there. If he continues to drink, then he will start finding a way after the money runs out.
My AH was sober 1 1/2 years then recently started again after starting to play softball and being unable to resist a cold beer after the game. He went to AA for only 6 months. But he is quickly able to see that his drinking is not under his control. He admitted to me that he feels anxiety when he's drinking and fears the drinks will run out. This week he went to some AA meetings.
Doesn't sound like your bf is ready yet. As I told my niece, please get informed so that you know what you are facing. My AH too was a peaceful drunk, but a drunk nonetheless. And I'm not sure you really want to start a relationship with this kind of start. And also beware that even if they may be sober, it's never over; they must work at this for the rest of their lives.