Hey there Welcome to the forum!
A Fiat Panda classic 750 in the USA! that must be awesome!
have you cleaned up any of the earthing points in the engine bay at all? follow the black wires till they go into a multiple "hedgehog"/spade connector tree, just give them all a bit of a fine sand papering followed by a dab of copper conductive grease on each one. that fixes pretty much all electrical faults.
there's one earthing point near the battery just below the suspension strut, there's one behind the headlamp where the washer bottle is, and there's another behind the.. (for a left hand drive car) drivers side rear tail light cluster.
one other thing to check would be the little flasher unit on the fuse box. maybe the load on it is too much when all 6 indicator/turn signal bulbs are flashing...?
as i remember it is the.. umm.. 2nd one across just above the fuses.
they can be hard to find .. a standard 3 pin flasher will work (so long as the pins are in the right order) but you may loose your dashboard light.. hence in the following picture there being a wire coming out of the relay going to the light in the dash:
basically i soldered a wire onto the coil side of the relay inside the flasher unit. so i still get the one dash light work for both sides.
as i remember, that flasher unit came from out of a Ford Sierra..
also if you just cant find one that will fit (the pins are all in the wrong place) on the fuse box its self, you can unclip the whole thing, turn it around and , by jamming a very small narrow flat blade screw driver down the side of the spade connectors, get them to come out so you can put them in any order you want. the whole fuse box just slides back in
if you have any pictures of the panda, i think i speak for most in the classic panda section here, we would all love to see it!
i only know of one other Fiat Panda classic in the USA and it is a grey 4x4 known as Juppy that now lives in a motor museum.
if you google Juppy the Panda you will find pictures of it
Edit: here it is parked up at Times Square: