I would get a wiring diagram and a trouble or test light and start either tracing the power from the source or from the end working towards the source. Not a fun job but know of no secret if you have already checked all the connections and grounds.
Before testing ANYTHING, you may wish to confirm you have the correct amperage fuses in your TS circuit AND Flasher circuits in case any wrong wires get connected or are incorrect. Would hate to hear your efforts went up in flames.
Start at the TS Flasher unit (2-wires), one should have 12V with the key on, the other goes on directly to the TS switch. You should be able to find the same color wire at the TS switch connection. That wire will be your "12V source" to your TS wiring and is the FIRST place to confirm. The TS flasher is simply a heated element interrupter. If you jumper the two wires at the TS flasher, your 12V signal going towards the TS switch should "stay on" constantly now (as long as the key is also "on"). If it is a bad flasher for example, by jumpering the flasher, connectors together, you will remove it from the possibilities of causes to your problem AND it will provide a constant 12V source at the TS switch to aide in any further testing (if needed) If this 12V source wire is connected to the correct 12v input wire of the TS switch, the rest of the testing should go fairly easily. If this doesn't get you going, you really must have a schematic at this point. Keep in mind that some replacement
TS switches have incorrect wiring colors so beware of this. In this situation, you will need to do a continuity test of the switch to determine what wire is the correct "source" wire for that switch.