You could possibly do one of the sinks that sits down and up. But connecting the plumbing drain will be difficult without removing the lower sink, or chipping around the existing drain to access the new one. Either way a risky project.

If I remove the lower sink, I can install a new one in the existing hole that could sit on the counter??

Highly unlikely that the holes would match. You would have to do a lot of planning, measuring, figuring out before you go that direction.

You may have to replace your faucet with a taller one, but you could install a shallow vessel sink over the existing hole (you'd probably have to remove the current sink to get the drain installed in the new one). This one is Length: 23-3/4" Width: 14"

This one might be too tall, but there are shallower ones available. There are also partially recessed vessel sinks, if you could find one with the recessed part small enough to fit in the hole.