I had a Logitech Extreme 3D Pro, never had any of the issues other people have associated with Logitech sticks. It lasted me for 3 years of daily playing X: Reunion, and about 2 months playing AH. I purchased a Saitek X52 HOTAS and the difference is night and day, but then thats comparing a HOTAS setup to a non-hotas setup, apples & oranges.
However, having said that ... since the cons seem to outweigh the pros, and since the cheapo Saitek stick has nearly as many functions, and is in the same price range, I'd go with the Saitek stick. Just to be safe.
To your questions:
Having a hard time keeping the crosshairs in place sounds like a problem with your mouse sensitivity, a cheapo joystick will most likely resolve this. Practice offline with the LCG (Lead Computed Gunsight ... Ctrl-Tab) and practice shooting the circling drones. This will give you an idea of the 'sight picture' needed to land hits.
This discussion will eventually lead to convergence and firing distances, but I'll leave those topics to those with more experience.