We often have the same problem of sopping wet lettuces. So as soon as we get them home, we wrap them in a lot of paper towels and put them loosely into plastic bags in the crisper. Every time we take them out to use some, we rewrap. Once the towels have dried, we can re-use them, so it's not as wasteful as it seems. And the lettuces keep ever so much better.
(All right, what's with this "we" stuff? It's really mostly my husband, because he's a real rabbit and eats -- and therefore makes -- the salad most. But he taught me that truc, and it really works!)