I actually went through this similar situation myself. In the end I asked two of my professors if the would write me a recommendation letter, and make three copies, each to different recipients. Neither one had a problem with it, and were very happy to do it for me.
Make sure you thank them properly for their efforts, though! I thanked both of them with some chocolates and potted tulips (it was around Easter,) as well as a hand written thank you letter! They were both very emotional and appreciative that I took the extra step to thank them!
Good luck!!
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