It looks like the part number you're asking about is 24350S which is a very hard to find and and an obsolete item. As far as lights not turning off, try to disconnect the wall unit and see if the light goes out without it. If it does, the wall unit is bad, not the sequencer. If not, maybe you're right and the relay is bad, in which case you will need to replace the relay. It's soldered onto the board, you will need de-soldering skills, a soldering iron and a solder suction tool to get it out.
Anyhow, pretty much everything you can do to troubleshoot the issue will require you to get up to the power unit and remove the cover at which point you should be able to simply lookup the part numbers of both the circuit as a whole (look for a sticker ontop of one of the relays)  and the relay as a replacement part if you go that route.
Be advised that the thing with relays is that in 50% of the time it's not the relay itself that's bad but the transistor that's driving its coil. If you can hear any clicking (anything, really) when you switch the lights on/off but the lights don't respond - the relay is a good candidate for replacement. If it's dead silent, it can still be the relay but can just as well be its transistor.
By the way, the replacement relay is Song Chuan 812H-1C-C SPDT (1C) 12A125V Coil=12V PCB pins which is available aplenty and cheap if you're able to wait 2-3 weeks for a shipment from China (eBay). At about 5 times the price there are a few local suppliers that would have it locally. Please just run a web search on the part number, they'll come up. Â Good luck!