yeeeouch. I'm not seeing the year of your bike, but these EX's are certainly ageing, and showing it.
Your observation probably correct...something lodged between needle and seat, could be worn needle too.
I'd urge you to order carb kit and have on hand, replace needles and all seals/orings while you're in there, also setting float height.
If you are able, a wet test prior to reinstalling the carbs is advisable. May save an R and R.
FWIW....a good idea as any carbed bike reaches 10 years old to perform a preemptive, full carb service, replacing all consumables, readjusting/renewing where necessary.
Custom Carb Service
**Being shot out of a cannon will always be better than being squeezed out of a tube. That is why God made fast motorcycles, Bubba.... Hunter S. Thompson**
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