Welcome to the forum
it would help if you were a litle clearer in your description as it is a bit ambiguous in places.
When you say that you connect the brown/white wire to the dark green wire I take it you mean within the switch yes ?
The dark green wire in the switch feeds the pulsed 12v+ from the relay, (brown/white wire) to the signal bulbs .The other side of the bulbs are grounded .
How exactly are you checking the continuity ?? you will not see a constant 12v+ supply on the brown/white wire at the switch because the relay should be pulsing the live feed
I would start by testing one side of the bikes signals ..... say the left side (dark green wire).
Remove one of the signal bulbs either front or rear so that it is not part of the test.
Verify that the remaining bulb works.
Then check continuity from the body of the bulb/holder to the frame of the bike to confirm that the bulb is grounded.
Then check continuity from the 12v+ end of the bulb (dark green wire) to the other end of the dark green wire in the signal switch.
If you still have continuity then repeat the test with the other bulb .
If all is well you have eliminated everything from the switch onwards for the left side signals.
If you repeat this process with the right side signals (Chocolate wire) then you know that the problem lies with the signal switch or before that.
this thread may help
http://www.xs650.com/threads/flasher-fault-troubleshoot-flowchart.42200/