yes that actually does sound familiar to me, does it also have rough engagement into a certain gear like 2nd? as in it takes a little more effort to push into the gate for that gear then normal? if so I would bet money your passenger motor mount has failed (under the coolant tank) this mount is notoriously weak on these models and fails rather annoyingly fast, the 1st gens (04-09s) where worse but these still do it.
all the motors (expect the speed) use the same rubber sections of the mounts from 04-13 the passenger bracket changed slightly from 04-09 to 10+ but the rubber inset part shows too be the same part (not the skyactiv or speed versions) to easily inspect that mount look at it, look for leaks that look like oil or power steering leaks but are right under and on that mount (it has hydraulic fluid in it) rips or tears in the rubber of the mount, or extremely soft rubber that collapses when you touch it (mine is currently in this stage and will need replacement soon, just kinda waiting till it gets worse, don't wanna use up my Saturday on this yet)
its easy to change though an air wrench or some strong might will help break the bolts free they are 17mm (all 4 the 2 nuts that mount it to the engine and 2 bolts that mount the u bracket to the frame) then if you buy just the insert and reuse the bracket the insert slides over a pole in the bracket and just rides there. then just bolt it back up, obviously you need to support the engine doing this a jack and a good block of wood help here, be careful and lift by the oil pan slowly using 2-3 people to help keep an eye on things helps as well. you don't need to lift much just enough to get the weight off the bolts/nuts and mount.
the drivers one is easy as well and should probably be done at the same time as the passenger (rubber rot and easier to jack the engine once) it is a "wishbone" type one under the engine (the one gearhd6 referred to right above me) the repair manual says to remove the air box and battery before doing this mount but I did it all underneath without removing anything else but the splash shield on my 08 2.3 and should be very similar on your car since the block and actual outer parts of the engine are fairly identical because the 2.0 2.3 and 2.5 are just bored and stroked different using the same block (though the drivers one rarely fails at lower miles, it is 7 years old now and rubber in the heat under the hood and outside in general will rot out especially at 7 years old) the passenger one absorbs most all the vibrations and shifting movement so a weak or fail mount there will make this more noticeable, there is a tranny mount but its stronger then the motor mounts seem to be and I never had any issues with it so it will probably be ok left alone
here is a youtube video that gives you an idea of where they should be (his is a 1st gen car but again uses same mounts so its the same thing) most of the time when you buy a new passenger ount you just get the rubber and have to reuse that bracket his broke out of (it just slides down and in) :
and I agree with gearhd6 the transmission moves way too much, what probably happened is your passenger failed and then the driver was doing all the work and over strained and has now failed too, so change them both.