It is not a relay it is a "flasher," and I don't care what some people call them.
They DO NOT have or need or should have a ground.
They are a SERIES device, that is, from battery.........to flasher.........through TS switch.....to bulb(s).......and bulb ground
What I'd do.........
You say you can swap bulbs. Do so. make CERTAIN bulbs are PROPERLY working on each side, and front rear. You can BYPASS the flasher with a jumper wire, into the two flasher connector terminals. This will feed full time power from the TS switch to the bulbs. Check CAREFULLY that only one filament of the two in the bulbs is lighting, and that the BIG filament is actually the one lighting.
This can get tricky because several simple problems can trick you up, such as........
1...Rusty damaged socket shell allowing bulb to fit with wrong contact orientation
2...Wrong type bulb
3...Worn/ damaged pigtail not making proper contact
4...lamp shell not grounding or not grounding well
5...Bulb with internal problems such as intermittent filament or cross--connection between filaments
Most/ all flashers depend on LOAD CURRENT (the bulbs) to operate. This in essence is the whole problem with LEDs, they draw very little current. Frankly I have not messed much with LED TS lamps but the key is certainly "the flasher" type and compatability.