My Peonies need Prozac

I don't know if I'm the only one, but I don't understand peonies.  I have zero experience with them prior to this spring and I had no idea what these weird things growing in my garden were until my mother-in-law clued me in. I think I'm still missing something.

OK, I get that the flowers are beautiful.... but seemingly only when cut.  As you can see from the picture below- the flowers right now are pavement dusters thanks to the crazy/crappy weather of the past week or so.  I have been able to rescue a few bunches and bestowed them on my friends.  Even prior to the weather, though, the plants themselves just looked like angsty teenagers- slouching and a little depressed.

The largest of the more than half-dozen in my yard (most are pretty skinny, small and barren of flowers) has a proper cage/stake with 2 rings holding it up.  I swear it started drooping more the second I gave it supports, though.  It was just pulling the supports over with it!  The next step was to attach the supports to a trellis that I stuck in the ground behind the plant. That's almost been pulled over too.

I just don't get it!  How can I make my peony plants themselves look as lovely as the flowers look cut??  If I can't figure it out, I may start giving away some of my peony plants for next year.  I don't want to rid the yard of them completely, however for a yard this size, so many drooping peonies aren't giving off the best vibe! (Do you see the pathetic scrawny one to its side?)

If you have experience with peonies, PLEASE leave me comments telling me what I'm doing wrong and how I can make my garden happy next year!