Signal lights won't blink, simply stay on

If the T/S switch isn't centered one brake light won't work.... Mopars of this vintage use the same filament for brake light as the T/S..

Dash lights check the dimmer switch, Front park lights are fed from the same fuse as rear park lights and the dash light fuse is fed from the same park light fuse...

The fact you changed the T/S flasher & it still doesn't flash points to bulbs not drawing enough current to cause the flasher to work... Are the bulbs LED's? If so they require a flasher that doesn't use the bulbs current demand as a way the trigger the flasher...