What causes the blinkers not to work is the fact that the "old fashion" flasher works on the fact that the bulbs draw a lot of amps, which causes the bi-metal strip to heat up, disconnect, cool, then connect again - causing the flashing result in the signals.
The reason LED bulb don't work is that the LED bulbs draw FAR LESS current and as a result they do not heat up the bi-metal strip so the flasher cannot work properly.
Replacing the (4) bulbs at the corners has no affect on the bulbs that you have in your dash for an indicator. You did state that you would also replace the flasher and install an "electronic solid state" flasher ( note that they are also polar specific - if it doesn't work, you'll need to swap the (+) and (-) leads going into the flasher ).
With the (4) corner bulbs swapped, and the flasher replaced, you should be up and running file. NOTE - got that exact setup on my 1966 Shoebox and the indicators are working fine in the vehicle.
Go for it - you'll love the results . . .