Sorry, I had a momentary brain outage - was thinking keyswitch. Like B+ said, a schematic would be helpful. My L35 is probably very similar, but this is the one area where my WSM doesn't match my reality - apparently the older L35's didn't have turn signals so the switch in the schematic only has 6 wires but mine has 8, if I recall.
Unless Kubota has changed things since 2004ish, the lights turn off with the keyswitch, unlike a car. The keyswitch feeds the comb switch. The only lights that operate with the key off are the hazard flashers, which are separate, I think. So I'm surprised you have 12v at the comb switch unless the ignition switch is on. This is one time that a test light would be better. The sensitivity of the multimeter will show power even when it may be just leakage through a poor connection.
So, as B+ has already said, check for dirty connections, using a test light (with a bulb, not an LED). Could be a loose fuse.