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I had this exact same problem with my GTS along with bogging and lower RPM's, Horrible gas mileage, Unsteady Idle if engine was revved, And the car was failing emmisions like it was the cool thing to do. Turned out the map sensor had gone haywire, Go visit a junkyard and pull one or something and try replacing it and see if it fixes the problem. I ended up figuring this out after replacing the ECU with three different ones, new spark plugs and wires, 02 sensor, air filter and a bunch of other crap the problem only stopped once the map sensor was replaced... As for the shifter moving forward and backward when gas is applied and released its natural in most manual transmissions its the gear tring to pop out of position when load is applied but its stopped by the syncros If there is signifiant freeplay and travel or it pops out of gear on its own during acceleration it may be time to invest in a tansmission rebuild.
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