If it's gonna sit more than a month, I either run it dry (hard to do without getting stranded) or put Stabil in the tank. The larger amounts in automotive tanks seems to last longer than a 3 gallon can for mowers, but it doesn't hurt, and if it does happen to sit there for a while, you're good up to a year & a half.
Sounds like you were pretty much on empty, then put fresh gas in it, so you should be good in that respect. Carbed, I'd check/rebuild it. I bet some varnish built up and/or has some junk in it.
The way you have to ease down on the gas tells me that it's starving for gas. The very FIRST thing is to verify you aren't out of gas. The gauge may be wrong. Fill 'er up! ^see below. Next, a fuel filter is a cheap thing to check. If you still have issues, pull the fuel line from the carb & put it in a small bottle of some sort & have someone try to crank it. If it "seems" like it's spitting it out good, get fancy & do a pressure test to be sure "spitting out good" is actually good. If it's barely spurting, you get to enjoy a fun job with the fuel pump. Carb cleaning, depending on your skill or aptitude. That should get you going.
I'd just finished a hardtail chopper & took off on a 1000 mile road trip. 50 miles in, start having trouble. I can start it, but any throttle kills it. This really stunk cause it was kickstart only. It'd idle for 5 or more minutes, though. I open the tank, I can SEE I have half a tank, and mileage/capacity said I should be good for 100+ miles. Like I said, I'd just barely finished bolting it all together. I didn't even have rear brakes. I expected to need to rebuild the carbs at some point, so I'd thrown a kit in my bags. So here I am, rebuilding the carbs and rejetting on the side of Hwy70 in a bazillion degree heat... no shade. Same thing. A few coarse words may have left my mouth. My riding partner put me on the back of her Honda Rebel 250 & we went up the road for a beer. It was sooo much fun arriving at a middle of nowhere honky tonk in the middle of the afternoon riding b1*^&. When we left, everyone actually came to the door to get a good look :P Get back to the bike, and I immediately knew the problem. When the rear got cut off & welded back up, it changed the angle of the backbone, and my tank was sitting at a 45deg angle. The petcock was smack dab in the middle of the tank. So even with a half tank of gas, I was... out of gas. I unbolted the rear mounts, rotated the tank back to horizontal, and propped it up with an empty Mtn Dew bottle. For the rest of the trip, that empty bottle sat in my saddlebag as a "reserve".
Anyway, the moral of the story is, even if you KNOW you've got plenty of gas, make sure you're not out of gas!