If they don't turn off, and you haven't changed the wiring from a known working configuration, then change the switch. Even if the light strings are non-dimming, they should at least turn off.

If you have changed the wiring configuration since it was last known to work, consult the switches manual. There were a few configurations possible to accommodate different lighting types. Also check that neutral isn't miswired somewhere and actually hot.

Those look like the current Lutron Caseta smart switches, which are LED compatible and according to manufacturer site may be compatible with low voltage lighting. Is it possible, and no offense intended here, that you're using the dimming buttons vs on/off buttons (the top and bottom ones)?

On the lighting... Those look very much like regular old rope lights which can be found on Amazon for about $1/ft

That link doesn't work.

Also, they're likely not 120v strands and/or the dimmer switch isn't LED compatible hence the wonky behavior.

I just had a similar issue where the end outlet read hot/neutral reversed (even though correctly wired at the outlet) and GFCI kept tripping. It was caused by mixing up the supply neutral (bottom) with the outgoing protected neutral (top).

Check supply hot/neutral are both on the bottom connectors and the protected run hot/neutral are on the top.