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:waving::waving:Welcome to the World's Greatest Goldwing Site gnis drnkr! :waving::waving:
A Guiness drinker eh? You're sure in the right place on this Irish based website.:cooldevil:
Not really enough information to make a good guess at your hard starting problem, it will take a bit more to determine whether it's a fuel or ignition problem. Start by checking your sparkplugs, do you get a good spark when cranking the engine? Do the spark plugs look dry after cranking for a few seconds?
The wobble is serious and the bike shouldn't be ridden until that's corrected. The most common causes of wobble on a 'Wing are a bad tire or loose head bearings. I'd give the tire a good going over to make sure it's seated properly on the rim, there's usually a molded in line near the edge of the bead. That line should be the same distance from the edge of the rim all around the tire and on both sides. Use a pencil or something and hold it against the fork tube just touching the tire. Rotate the front wheel and watch the gap, the tire shouldn't show any runout more than a 1/32 of an inch or so on either side. Do the same for the diameter, hold an indicator close to the face of the tread and turn the wheel to check for out of round. While the front wheel is off the ground slowly swing the handlebars from full left to full right feeling carefully for a 'notchy' feel. If there is any roughness or notchy feeling it's likely the head bearings need attention.
If those check out, with the bike still on the center stand and a jack under the front of the engine to keep the front wheel off the ground push and pull on the wheel. There should be no play front and backwards. You can also check with the bike on the ground and the transmission in neutral. Hold the front brake and rock the bike back and forth, you should feel no play in the forks. If you do it's possible you have loose head bearings and need to inspect and retorque them. If all checks out it might be a good idea to replace the front tire.
One more possibility would be the swingarm. With the bike on the centerstand you can try prying the swing arm sideways to see if there's any play. There should be none.
Whatever you do correct this problem before riding the bike again, something's seriously wrong.