If we're flying out late and want to enjoy the beach before we have to leave, is there a place that we can clean up or shower after we leave our room?

Yes, both on the arrival day before check-in time and on the last day after check-out time you can use all of the facilities.

There are showers in the locker rooms near the Cobia pool.

Yes, the welcome center has large bathrooms and there are other areas near the pool.