Shifting problems while coasting/accelerating
 

I have a 2001 sonata that has started having some shifting problems lately. The automatic transmission shifts great while accelerating and generally "feels" okay decelerating. However if I am at speed (say 45mph+) and I take my foot off the accelerator, the transmission seems to disengage. I say that only because when I depress the accelerator again, the transmission seems to not be in gear. If I press slowly, then the transmission will gently engage and then accelerate. I see the rpms increase until the transmission engages and then it shifts through the gears fine from there. If I accelerate quickly, I would feel the car jerk as the rpms quickly rise and the transmission engage. Alternately, if I keep gentle pressure on the gas pedal while coasting, the transmission stays engaged and operates completely normal. Any thoughts on what may be the source of the problem?

How many miles since the transmission was serviced? I would start by simply checking the fluid level and condition of the fluid (brown color, burnt smell, etc). I suspect bad fluid first rather than low fluid from the symptoms you describe. If not fluid related, what you describe sounds mostly like the torque converter is dying to me, which is a rebuild situation.