There are two things to know that maybe (hopefully?) will make you feel at least moderately better about letters of rec:
1. Professors do genuinely view this as part of their job. This is not a favor they are doing for you - this is part of what they are being paid for. I'm an instructor, not a tenure track professor, and even I view writing letters of rec as part of my job. It's never a burden. In fact, I had one student ask me for a letter with a one day turn-around because he just found out about a scholarship, and I got it done. It's karma for all the letters we've had written for us.
2. I assure you that your writers are not writing new letters every time. They are pulling up the file from the last time they wrote it, tinkering with it for a bit and then sending it on. Once you've requested a letter from them the first time, the subsequent requests are probably taking them 10 minutes, tops. Again, you're not a burden.