Do university know/care if I've dropped out of two other university courses

When applying through UCAS you need to declare your prior study at university - failure to do so can lead to your application being flagged as fraudulent, any unis that have made you offers are free to nullify those, and UCAS can exclude you from the remainder of that and potentially even further future application cycles.

So, since you'll be declaring it, the university you're applying to will know. Whether it will affect things is hard to say, in my experience it didn't but I had a gap between my original studies and my new ones of a few years where I was working. Generally a false start on another course is usually not something unis have a huge issue with I gather, although two such false starts may be a concern.

However, the bigger factor is if you've now left two courses in a row, you won't get full funding for any new full time course you start from year 1 now, due to having used up two of your years of funding entitlement, unless you apply and are granted an additional year of funding due to compelling personal reasons (CPR). You should speak with SFE and understand what the implications are and what your options are urgently, so you go into this with your eyes open.

Note also that if you have to self fund a year of tuition fees due to not having enough fee entitlement to cover the remainder of the course, that will always be the first year(s) in the new course, not the last ones. So you can't plan to work and save money for the first two years to self-fund the third year - you would need to be self funding the first year of the new course.

That said, something to consider also is that part-time funding is a separate process and if you haven't earned any qualifications, your prior full time funding shouldn't affect your entitlement for part-time funding - so you should still be able to fully fund a part time degree