When you say they go on and won't turn off, I assume what's meant is that they don't blink? If the signal lights go on with the switch, that's 95% of the work done already. The flasher unit is not cycling, so it is either defective, or more likely, it requires more current than the installed bulbs draw to cycle - particularly if they are small aftermarket or LED lights, or it requires more current than is available from the battery when the bike isn't running. The flasher relay has to get it's rated power to flow through and thermally cycle, unlike the late electronic ones.
If you mean they go on and then cannot be turned off, that's a little different and not the fault of the relay, but more likely the switch itself or, as you say, an underlying wring fault. This seems less likely.
The Lectron Carb was popular for racing bikes for a time in the early to mid '70's, but they are an iffy choice for streetability.
