Hello, I can relate as I am applying to a total of 9 different programs, and I'm also applying for fellowships at almost every school AND outside fellowships. I've used a group of 6 different references, but two professors have done most of the work. I've found that asking that the following has helped when asking recommenders:
1. Ask EARLY - give them at least 6 weeks, they'll probably get it done well before then.
2. Offer to help in any way you can - for one reference who is a busy middle school teacher, I actually re-typed a letter that she had previously written for me (she had lost the doc), tailored it to the school and gave it back to her in a hard copy to sign, and emailed the file in case she wanted to change it. In no way did I "doctor" it, but I saved her a lot of time.
3. If the school gives you the option, paper letters are sometimes less work (unless they have a form that entails more than the signature).
I always OFFER to make arrangements in case they prefer to send a paper copy of the letter, but in my experience everyone was fine with the online app.
4. Remember that those who agree to serve as references believe in you and your abilities, and although it's tedious, this can also be a fun part of their jobs. You're worth the effort:-)
Good luck!