My derm suggested Cereve and it is awesome. It's not particularly foamy, but it makes my skin feel wonderful.
This. You might not like it since it doesn't foam, but it's a great face wash. I'm not sure it will give you the squeaky clean feeling, though.
Something else that doesn't foam, but you might like if that isn't a dealbreaker, is Lush Angels on Bare Skin. It is a very odd texture for a face wash, and they cut it off of a big bar (like their soaps, if you are familiar with Lush). You then pinch a piece off, rub it in your hands with a bit of water, and then rub onto your face. It is very gritty, but this is great because it exfoliates really well. It always left my skin feeling very clean (that squeaky clean feeling you are going for) and also soft.
I know I mentioned this above, but I'm REALLY loving this Origins Never a Dull Moment wash. On top of all the stuff I mentioned before, I was looking at my face today prepping for a masque, and discovered that just in the two or so weeks I've been using it, my PIH marks are GREATLY diminished.
I use Neutrogena liquid face wash (just the basic orange one in a pump bottle). However I've found that the oily nature of my skin is a direct result of the moisturizer I use/don't use, rather than the face wash.