Can I trade my car that's still owned by the bank? If so, what are the consequences?

The dealership (if willing to do so) will take the trade in, pay off the loan entirely, and tack the extra owed onto your new car loan. For example, using the figures you posted above, you would owe the difference of $4K (23.5 - 19.5) so if you then financed a car for $20K, you would really have to finance $24K (20 + 4)