When I first took guitar lessons in school, we used and played mainly classical stuff so I got used to strumming and fingerpicking with my hand. Now I find that I really do want to focus on not using a pick to strum or pick.

With chords, it seems more difficult. I can handle most open chords fine:

C (x32013)
F (xx3211)
G (320033)

The main problem is with upstrokes on something like the "F", I'm not exactly sure of the best way to handle that (I use the back of my thumbnail).
Well, for an upstroke I suggest you use your thumb nail. Not the skin, that gives it a sorta...dampered sound. You can grow your nail out a little bit if that would help you, I keep mine moderatly long.

On the barred chords, your problem is probably not the picking so much as the chord its self. Make sure you're putting enough pressure on the string as to be sure that all the strings are making a clear sound. Try holding down and F and picking each of the strings individually. Do they all make a clear tone? If not, keep working until it does. That might fix your problem.

If none of that helps, then you probably just need more pratice. It does take a bit of time for most people to switch from a pick to fingers but hopefully not too long. Just give it a little bit of time.