Is it alright to ask recommendation letter from the same professor more than once?
tl;dr: yes, but do explain the circumstances.
Professors are well aware that writing recommendation letters is part of their job. We know that good letters are a requirement for students to advance, and if students have behaved reasonably well in their interactions with us, we should gladly provide such letters. I had a student apologize to me for asking me to send five letters to distinct employers, so I can understand your reflex to not want to be a burden. Let me reassure you, just as I reassured that student: it is not a burden at all. Writing recommendation letters is a bit of work that results from having had interactions with the student that I enjoyed.
All that being said: you probably will want to make sure that the professors will understand that their work has not been in vain. Thank the professors for their efforts so far, express that you are very happy with the achieved results, and tell them that you unfortunately must ask for a little more of their time. Then write the reasons why you couldn't accept any of the spots last year, and ask them to send the same letters again (only updating the cosmetic stuff, such as the date).
It may be difficult to be open about your personal circumstances to the professors. However, it is probably necessary to tell them why you couldn't make use of the opportunities last round, since this motivates why the professors wouldn't be wasting their time this round. If you write it like that, I would find this request absolutely appropriate, and I would be surprised if your professors wouldn't respond positively.
Good luck with the applications!