I haven't had that problem with Recollections Cling Stamps--The problem I have with the Recollections cling stamps is that when I peel them off my storage sheet, it leaves behind parts of the picture that is on the cling foam...but that's not a big deal really...
I do have that problem with a couple of my Hero Arts Cling stamp---the ones with a vinyl-like material on the other side of the rubber. They fall of right when I'm about to stamp
and also when I store them they do not stay on my plastic storage sheet!
I wrote Hero Arts explaining the problem and asking for any ideas on what to do---I too have washed them with soap; made sure all lint was off of them by applying gift-wrapping tape on the cling and pulling it off over and over to get any lint off---nothing has helped----but sadly Hero Arts never responded.
The only things I have been able to do is either stamp it by putting the paper to the stamp while the stamp lays on the table----I have trouble with this method, though I know you have told me you have success with it in the past with large stamps. But one of my Hero Arts cling stamps is kinda curled so not sure this would help...it's like the vinyl backing has SHRUNK on the curled one!
The MAIN thing I do with this problem is use a product I use with my unmounted stamps and those I have carved---Daria mentioned this product to me and I love it! It's Tack 'N Peel by tsukineko:
link for Tack 'n peel Tack 'N Peel is very tacky and anything I have ever stuck to it has stayed no matter what! You get 2 sheets of it and I have only used HALF of one sheet---I cut the sheet in half and stuck it to one of my 6x4 blocks and have left it for almost 2 years now. I just leave this product on one of my acrylic blocks all the time and if I need that block for another stamp, then I use the other side of the block the Tack 'N peel is not on. When I am not using th Tack 'N Peel side, I leave the plastic cover that covers it on it. It's still tacky---if it gets less tacky you just wash it or apply tape all over it get any lint/dirt off--then it's just like new.
One other idea I had was maybe applying heat to the vinyl on my problem stamps and see if that helps them stick before applying them to blocks...haven't tried it.
HTH!