Right turn signals stay on, left stay off.

you should be able to drain your carburettas without removing them. look for a drain screw on the bottom of the carby bowel. If it is fuel injected, then you may have to replace the injectors.

Try spraying inside of the switches with WD40 or similar and switching from one side to the other rapidly for a minute. Contacts on switch possibly slightly corroded due to damp.

Hi!
From your good description I would say either a bad earth or corrosion.
Check bulbs too as wrong wattage can cause grief.
Make sure all bulb contacts inside holder are not corroded (With ignition OFF) also the sides are clean too for good earthing, still a problem? then check all wire connections to each indicator, a little WD40 or similar all round is a great help when doing this.
Good Luck!
Paul 'W'
Onyer~EDson(:0) {#><U.K.

Make sure and replace the lamp with the exact same one as the working side. Sounds like their is a mismatch in the lamps of some sort.
If one of the filliments is burnt out this will happen. Just like in car blinkers. Or it will blink fast. Hope this helps.. Happy Holidays. Jay