old home. no ground wire
No. "Romex" is a trade name and is actually a brand of non-metallic cable or NM cable. Article 334 of the National Electrical Code.

If your house was wired with NM cable in the 50's or before, it most likely has no ground wire.

Smart switches require either an equipment ground or a neutral wire to power the smart switch while it is waiting to turn on the lights. You should not depend on visualizing wires in the boxes. You should test your system to determine if there is an equipment ground available at these locations. There was a method in the 50's of taking the equipment ground and wrapping it around the cable and then clamping it into the box. So, you boxes may be grounded after all.

If you don't have an equipment ground, all is not lost. You can run separate equipment ground wires to any of the boxes where you need one. It will be a hassle but the switches will not work without it.

Good luck!