There are so many variables involved with the early turn signals/Hazards, they can be quite difficult to troubleshoot.
Start with the basics.
If ANY of your turn signal lamps are LED you MUST use an LED flasher!!! That fast flashing I hear could be a sign of LED bulbs installed somewhere or it could be a sign of a bad flasher. First thing I would do would is to swap in a know good flasher
Next I would pull the Hazard switch, take it apart, clean all the contacts, and re-lube it. When reassembling make SURE you get the metal tabs holding the phenolic contact board down tight, or you will have intermittent issues. I use a vice to crimp those tabs down tight. Dirty contacts in the Hazard switch can cause all kinds of weird issues.
Next I would verify that you have the new turn signal switch connected to the harness properly. It is VERY easy to get the turn switch bullets connected to the wrong wires on the harness side. Toyota knew this and provided a means to identify the wires easily.
The shorter two wires (Green/Yellow and Green/Orange) coming out of the OEM turn switch are for the FRONT turn signals. They plug into the triple bullet connectors on the harness side along with the dash turn lights. The longer two wires (Green/Yellow and Green/Orange) on the turn switch are for the rear lights.
Again, the triple female bullet connectors on the harness side are for the front turn signals and the turn signal dash lights.
Then I would check to make sure you have the right bulbs installed. Since the flasher is a current activated device, it has to have a certain anount of current flow before it will flash. Using the wrong type bulbs (ones with less current flow ie dimmer bulbs) can cause the flasher to not activate. Make sure the two sides are running the same bulbs front to rear.
Grounds
Since the 1970 lights rely on a basic metal to metal method of grounding, if they are rusty you will have an issue with current flow resulting in dim lights and/or no flashing. I would HIGHLY recommend you run a separate 16ga White/Black ground wire to each light housing and ground it to the frame or to the body. This ensures you have a proper ground for the lights. Also check for corrosion in the bulb sockets themselves. A Dremel tool and a wire wheel can clean those up in no time. Note: the dash turn indicator lamp sockets ground to the dash through metal to metal contact. Make sure they have good contact with the dash. You can also solder a ground wire to the lamp socket and run that to a bolt under the dash for a better ground for these two lights.
Fuse Blowing When activating Hazard Lights
If your rear chassis harness is mangled, you may have a short in the rear harness. To test this unplug the rear chassis harness from the main harness under the hood at the passenger side firewall. Then test the hazard lights again. If they work then you know the issue is in the rear harness. If it still blows the fuse then the issue is in the front wiring somewhere.