Many great suggestions in this thread. Here's my $0.02 (US):
Depending on the song, you may even be able to shorten the chord form from the posted "cheats". If you're picking more on the high strings (G-B-E), go ahead and ignore the low strings (E-A-D). That makes F look like xxx211 and Bm look like xxx432. You don't get much (any?) bass sound, but if you're already on the treble side anyway, it won't make that much difference.
Another option: change to a different chord altogether. This works better if you're writing a song yourself than playing someone else's, but every now and then you can work something in to give an established song a little different sound, and make it your own (and easier to play). For example, when I started out and had problems with F, I sometimes changed it to Fmaj7 (xx3210) instead. Different sound, but it can keep you moving with the song while you improve on the barre. I did that with a song I was writing because I just couldn't get the barre to sound like anything, and now I like the sound of it better in that song anyway.
Last, and I hate to sound like everyone else, but practice improves play. When you're at home alone, go ahead and play that F with three dead strings in it every time. Every time. Over and over and over again. Soon enough, you'll only have two dead strings, then one. I won't guarantee you any better than that, because that's still where I am most of the time. My boss plays professionally and says that he still routinely hits 5 out of 6, and sometimes less than that. At least we're in good company.