Just trying to start my 1978 1/2 FLH for the first time this season. Usually, after it's been sitting awhile, it won't start unless I remove the spark plugs and splash a little gas in the cylinders. That's not working this year. It only runs until the splashed gas is burned.

I took the carburetor cover off, and it's wet with gas there. When I pull the throttle, I don't see any gas spraying. I'm wondering if I should see some or if there's something else I can do to get this bike running.

 

pull your gas line off and drain the old gas and that will give you a good idea if you have good flow thru screen,if not good flow, pull the petcock and clean screen.
now fresh gas and good flow ? if than doesn't get her running pull carb and pull float bowl, check needle valve to make shur its not stuck, squirt some wd40 to lube it up or set carb in a can of gas over night, some times it will help break the crap loose if pluged, pull your intermediate and main jet,blow with air pressure .
sounds like punk gas

 

Petcock screen needs cleaning and probably the fuel & cross over lines need replacing . You may also find that the cross over tubes coming out of the tank are plugged up too . Wouldn't hurt to check the vent side gas cap also, those will crude up and it's just like shutting the gas off