Changing the fuel tank and hoses (helped it to work for a while and then went back to the same problem)
I have two of these motors, a 1998 and a 2001. They are very good, and run well, and are repairable. If you give up, I'll buy yours.
First, the OEM fuel tank is garbage. I had one fail because the clear, vinyl-like pick-up tube shrunk and became loose, probably because of ethanol in the fuel. By 'shrunk,' I mean the wall thickness decreased, and it no longer sealed. I had symptoms like yours. I bought a new fuel pick up assembly for $26, and it worked for a little while until the same thing happened. The only thing that fixed this was buying a N.O.S. metal tank on eBay. I still use metal tanks, and no problems in this department.
I have had one electronic module fail. Replacing it was easy, and fixed my running problems.
There is no choke on this carb. It has an integral fuel pump, and a primer. The fuel pump diaphragm and primer parts come with the rebuild kit. A complete new carb $214.41 in 20014. I bought one because I broke one tightening it too hard.
Throw away the Merc tank. Get a metal tank. If problems persist, consider an electronic module ($162.03 in 2015). Maybe coils, but I have never had a problem with them.
Good luck.
p.s. check and change the lower unit oil every year. If it's milky, stop using it, flush it, and rebuild it with new seals. If you ruin the lower unit your options are very limited. It is cost prohibitive for anyone to rebuild it, and a new one is almost as much as a new motor ($1400).