I just participated in a similar discussion on another board–the general consensus was to cut your losses and toss them. 

If you are too frugal to do that, adding lemon juice might help, and some folks said roasting concentrated the flavor. 

You could roast them and turn them into freezer jam. 

Pete said:

Mushy fruit always reminds me of fruit butters.  We had some less than tasty peaches that we turned into a quite wonderful peach butter earlier this summer. 

I was going to say the exact same thing.  Get outta my head, Pete!  If you puree them, and cook them down a bit to concentrate the flavor, I bet you'd get some really good peach butter.

Oh, fruit butter ditto, yes, oh yes and yum!  I take near every fruit that ripens in my area and do a butter for each.  It's my first go-to for any fruit slightly overripe or unimpressive.  Butters do reduce to near half of what you start with but what lovely concentrated sweetness with low extra sugar and no pectin!  I fill my favorite 5-qt pot with coarsely chopped fruit and pour in one container of condensed apple juice or condensed white grape juice depending on the fruit, maybe a slosh of a wine.  Cook till soft and render silky with an immersible blender.  My last butter was 16 pounds of pears reduced with a half cup of white wine, one can of white grape, 2C cane sugar, couple tsps of ginger.  My closest friend lays in wait for this put-up at the holidays, declaring it "liquid pear" and eats it with a spoon straight from the jar.  I think your soft peaches are a perfect candidate for a fruit butter attempt!