I'm not sure but the 82 model CB650 I owned had a set of very fine petrol filter screens under the float valves, (they are the wee pointy things that the floats (brown square things) are attached to) once you remove the pivot pin (axle) you can lift away the floats and just check if there is a set of screens in there.
I would guess that as your bike sat for 5 years with gas in it, the float valves got so used to being in one place and with a build up of crystals and petro-chemicals they need a decent clean, just clean the with a clean rag and push the wee tips at the end in and out a few times to make sure the tiny springs are not stuck.
After that I can't see why all your carbs would leak at all, the float height is probably correct as when you got the bike it wasn't spewing gas out of them eh?
Once you drained them and added new gas you probably created a whole new set of problems.
Edit: Don't under any circumstances play with or try to remove the grey slotted screws at the ends of the bowls, they are set when tuned and if you play with them you will need a bit more help from the forum.
You have the one's that were made specifically to be "eco-friendly" so the nice people at Honda fitted them with a very small range of adjustment so you could not enrichen them and pollute the earth.