You may find that the 2006 circuit diagram is going to be fairly close.
http://mog2904.free.fr/morgan-traditional-parts-book/morgan-traditional-parts-book-2006.pdfThe fact that the flasher unit is faster than normal means it is an electromechanical unit so I would start there.
Do both sides work when you use the hazards? If so them the fault is probably from the flasher back to the stalk switch.
With the ignition off you should be able to continuity check (with a meter) back from the stalk switches to the fuse box and the flasher unit and then to each bulb.
Try and find the flasher unit part number and get the connection diagram from the web, they are usually fairly common pinouts. Make a drawing on paper as you buzz out the circuit (you may need to use a long wire with a crock clip to extend your meter).
You are looking for a faulty flasher unit or stalk switch or bad connections (inc earth) at any point.
With the ignition on you should be able to measure the flasher unit with the meter (on volts) and see if the flasher unit is working.