If you live in the US there is FAFSA grants to go to community college for free. Or also, if you can get into a trade, construction, or commercial drivers license (driving trucks, etc..) Or perhaps work your way up the ladder at UPS, which is unionized, good pay, etc..
If not living with your GF means you are screwed eitherway, I'd say, just earn your keep at your girls place best you can, forgive and forget to the best of your ability, and bide your time working towards a more sustainable income.
There's plenty of, 'bipolar,' people who don't need to cheat. The majority of the population has, 'self-destruction,' issues. That's a whole different can of worms though. In my experience, people like that, tend not to change. There's plenty of ways to hurt one's self, to distract from other pains, without it affecting others, or minimally affecting others. Smoking for example, is self-destructive; however, people deserve the right to indulge in a pleasure such as that, if they so desire.
Sounds like you are both still quite young, yet, you also have some heavy and mature responsibilities to consider.
Consider starting a meditation practice, as well. On youtube I think there is a free audio book of, 'zen mind: beginner's mind,' and a slew of other resources to learn meditation.
Mingyur Rinpoche is some one I find mostly agreeable, myself. As a beginner you can try 15 minutes/day, and then try to work up to 30 mins/day. I do 30 mins a day, which is probably the minimum. Some suggest at least 1 hour a day, should be the minimum. I'm not really sure that, 'how much,' matters so much, quality over quantity perhaps; however, being able to center one's self, and tap into a more relaxed, observant, aware state of consciousness from time to time, can help with the stressors of life, in general. You can also see if there are any Buddhist temples in your area, or Church's, or whatever suits you. (I tend to go it alone myself, but, it can be a great boon to connect with others.)
So, consider Community College (certifications, etc..), FAFSA, trades, commercial driver's license, construction, UPS, etc.. Bide your time, and work towards finding a better paying job. Let bygones be bygones and accept that, it's unlikely you will be able to change another person, or see them change; change often begins with one's self. And perhaps take up a meditation practice, or yoga, or some sort thing like this (it can help to take refuge in a daily practice that is beneficial for both mind and body). And then, just rationally measure the pros and cons: con: she cheated on me. Pro: I have a place to live and perhaps from that position, I can increase my ability to provide for myself and my loved ones. Make the most of the positives, and learn to manage the negatives.
Anyway, about the best advice I can give.
May it be well with you.