i have an automatic 99 hyundai tiburon i'm trying to fix up.

symptom 1: my steering wheel starts at 12:00 center. as i accelerate anywhere faster than "granny" speed my steering wheel center now becomes 11:00. when i let my car lose G-force, as i approach my planned top speed, the center very quickly and harshly returns to a normal 12:00. this will throw me left into oncoming traffic if i am not anticipating the sudden steering wheel jerk.

symptom 2: my steering wheel starts at 12:00 center. as i take a right turn with any noticeable amount of G-force, an example of i am turning the steering wheel myself from 12:00 to 1:30 around 45mph, the steering wheel center becomes 11:00. this makes my 1:30 right turn at 45mph a 2:30 right turn until i slow down or lose the G-force. this symptom is the scary one as it's not something that can be anticipated. i just have to take right turns slowly. highway travel at speed limit is impossible with any sort of right turns causing G-force.

i am mechanical but not a mechanic. the only bit i know of steering is what i've been looking into.

this is what i have diagnosed wrong so far.
* outer tie rod end passenger side shakes slightly and needs to be replaced.
* the axle on the passenger side has a broken axle bearing resting on it. the boot is untorn and in decent condition. i'm guessing someone replaced an axle bearing before and didn't bother to take the broken one off the axle.
* the control arm inner bearing driver side is loose. the rubber is torn and cracked. the nut freely spins. needs to be replaced.
* both half-shafts of the axle have a very slight amount of play.

i will be fixing all i have noticed but am still not sure this is going to solve my issue. the control arm bearing might do it, but i'm hoping someone can confirm or point me where to look.

if by shimmy you mean, wander, then no. outside of symptom 1: where i am accelerating in a straight line.

the steering wheel itself feels like it should. stiff, but easy to turn.
the car actually drives very well. it's just the switching of steering wheel center during faster acceleration and moderate right turns.