so you are getting juice to both pins in the fuse box with the flasher out? hopefully if im wrong someone will correct me but it should only be 1 with juice when signal lever is switched on ( either way ) and the other is the output to the lights.. without a flasher the 2nd one should not be live.. also the flasher is a thermal style.. which once so much electric runs through it it overheats and shuts off, cools and kicks back on.. of course doing that at a fairly fast pace..
the issue seems to be in the wiring coming out of the fuse box going to the lights themselves.... its a dumb question to ask.. but do you have the proper bulb in it?? not trying to imply anything but i know a wrong bulb can wreak all kinds of havoc... it should be a 3157.. the dual filament bulb...
but if not it sounds as if somewhere you are getting a crossed wire and supplying juice when it shouldnt.. check to see if they are on and lite when they shouldnt be ( i.e. marker light setting but turn signal brightness... ignition on and light come on.. all time live ) but first take a peek at your bulbs just to be safe