Whatever fell down the sink drain, is probably not the cause of water backing up in the shower pan. That thing is most likely still stuck in the "P" trap under the sink.
I would suspect a blockage in the grey tank vent line. If that is plugged, the tank can only take so much water, until it gets pressurized. Then the water takes the easiest path available. That path would be your floor drain.
The vent line, in my TC, is a 1/2 inch clear plastic tube, attached to the grey tank, near the top. It then runs up above, where it ties into the main vent. In the past, that line has slipped down below the connection point, and when the tank got full, or close to full, water would get into that line. Once the line was blocked, I would get backup in the floor drain.
Couple ways to clear it. (a) disconnect the line and drain the water. (b) Open the dump valve (into a bucket) and lift the line to drain into the tank.
I used a zip tie to hold the vent line from sliding down and forming a trap. No problem since.
Look for an access panel or door, to view under your floor pan.
While you're taking things apart, might be a good time to clean out the "P" traps under both, the sink and the floor pan. After cleaning, pour a cup or two of water in the drains, to re- fill the traps.