A slightly worn and leaking accel pump diaphram cause your symptoms. Try that first. 3 screws under the carb, right side (front) and replace it with an HD OEM one, not aftermarket.
If not, it sounds as if the Choke pintle passage and pintle itself is starting to get carboned up and needs to be cleaned. Choke cable beeds to be taken out of carb, and the pintle needs to be cleaned. In order to clean out the port, you need to take the carb off, unless you can speak a can of 2+2 with the little red hose thingy back there, and spray it in the hole well enough. It can be done of you just take the carb out of the intake and tilt it down.
Truthfully....if you're going to that trouble, I'd rebuild it while off and clean all the other passages as well. There's 12 years on that carb, and that's if it was rebuilt the day before you picked it up.
If that works....yer golden. If not, prepare to rebuild.
“When the going gets weird, the weird turn professional....” ~ Hunter S. Thompson~
R.I.P. Rubberdown. 1971-2012
"Your Jedi Mind tricks will not work on me...... ~Misinlink-2016~
2005 E.G. Classic
S&S 510 cams
Baisley Oil pump spring
SE 3.5" pipes
S&S adj pushrods