If you have Picasa, you have Google Drive - they're one in the same now, same storage space. I guess there's no reason you couldn't create additional free accounts, and use one for Picasa and one for Google Drive.
Dropbox's free account is pretty limited, 1 or 2gb I think. Their Windows client does allow for super easy sharing thought.
OneDrive used to be very generous, giving you 15GB for free, however they have limited that to 1 or 2gb now.
You mention your Picasa storage gets full? Are you uploading HUGE pictures? If you limit the size of your uploads, your photo storage with Google is "unlimited", and does not count against your 5gb of "free" space for Google Drive, for sharing non-photo documents.
Just FYI, if you subscribe to Amazon Prime, you also have "unlimited" photo storage through Amazon Photos, restricted by size of the photo.