Something doesn't sound right there, because contacts shouldn't count towards your Google Drive storage limit, and it seems odd that all of your devices would stop syncing everything (rather than just Gmail, Google Drive, and Google Photos) because of your Google Drive limit being reached.
In any case, if you turn sync on, I believe all new contacts will be added to your Google Contacts in the cloud, and therefore synced to all devices that are logged into that same Google account. It might be a little trickier if you deleted a contact on any of the devices -- if that contact had already been synced to the cloud, then deleting it on one device that then syncs with the cloud would probably delete it from all connected devices.