Your problem sounds like a bad connection or a combination of weak connections. Your symptoms really point to a bad contact in the turn signal switch.
I am speaking more from working on Jeeps and Broncos, so take it for what it is worth. A deuce flasher system is identical to Jeeps and Broncos from what I have seen so far, except for the voltage and those COOL double element bulbs.
American electrical systems use beryllium-copper for contacts. It makes them less corrosive and springier. If you want to see why that is useful, work on an ‘80s or ‘90s Mercedes electrical system that uses pure copper! Beryllium-copper will still oxidize to some extent and will eventually break the connection. Two contacts firmly pressed together can loose contact after about 25 years.
Also the problem is cumulative, meaning that ALL CONNECTIONS INCLUDING GROUND should be opened and cleaned. A ¼ inch drill with a 17 or 22 caliber bore brush is great for getting down into tight connectors. Also do a thorough inspection of all wires. A cracked or pinched wire will let salt in and the wire can be significantly degraded. Crimped terminators can be making poor contact for the same reason. The solution there is to buy acid-core solder and solder over all crimped contacts. Don’t open them, just solder over them.
You say that you put on a new turn signal switch but it could still be at fault, especially if it was a pull-out. It could have the same problem, even though it was working in the other vehicle. I recently bought a “re-pop” (brand new 3rd party manufactured) turn signal switch for my Bronco that had exactly the symptom that you are describing. I had to re-work it because the contacts didn’t quite make.
You can confirm the switch’s operation with an ohm meter. On a single contact with no more than 6’ of wire, you should not see more than 0.2 ohms. Remember to measure the resistance of your leads first. If none of this works, get a diagram (someone on this link will know where to find it) and trace it out. Watch for excessive resistance.
I like cranetruck’s solution. Early GM starter systems, which Jeep purchased, ran the starter solenoid current, several amps, up through the ignition switch. It worked fine for the first 10 years of the vehicle's life but would drive you crazy after that. A Ford starter relay mounted on the fender was the solution. Cutting down the amount of current that you are running through the switch can be very helpful, especially when you pull trailers.
Sorry, had to throw that one in there.
