The turn signal console light can be a pain till you figure out what's happening. As built it's just a light globe between the left & right sides. It draws power from whatever side is on and uses the other side as an earth, relying on the globe's high internal resistance to stop the "off" side from being lit up by the "on" side.
An LED has very low internal resistance, and will only allow power to flow in one direction (because an LED is a type of diode) so if you to a direct replacement it won't work at all in one direction, and both sides will light up in the other direction.
To make it work you need to do two things: first you have to add a proper earth wire so the LED doesn't have to earth through the "off" side. This is most easily done on our bikes with the tacho's earth connection. The earth gets connected to one side of the LED. Secondly you need to connect both the left and right to the other side of the LED so either can make it work, but if you just connect directly both sides will light up when you turn either side on, so you have to stop the left from feeding the right and vice versa with a pair of diodes (which act like one way valves for electricity).
Finally, if your LED doesn't already have one built in you need to add a resistor in line with the LED so it doesn't draw too much current and kill itself. Because you have an "LED replacement for an automotive globe" it'll already have the resistor so you won't need to worry about that, but you'll still need to do the rest.
It used to be that the bigger problem was the tail/brake light as it projected light both to the licence plate through the top of the globe, and out the back of the bike through the side of the globe. My solution was to build a circuit board with LED's facing in both directions that completely replaced the socket, but I see SuperBright now have an LED globe replacement that will do that for you. 3 years late, from my perspective but good for you.
The other thing to be careful of is not to replace your high beam indicator with too bright an LED. This light can be on for long periods when it's dark and will burn a hole in your retina if you go too bright.
