Hi my name is ***** ***** I will be happy to assist with this today.In some cases I may need more info to assist better.I apologize for any delay.
When stuck in the mud it usually takes a good amount of power to get the vehicle out.I have seen in several cases when getting a vehicle out of the mud that this can cause an issue with damaging components.The biggest issue is the transmisison,transmisison mount breaking or even drive line or drive line connection/joint damage.The exhaust may have mud in it causing the burst you see here and may need to be driven to clear out.
I would have the under carriage inspected fully for any damaged components needing repaired or replaced to correct this here.If any issues are found the affected components will need corrected.
I suspect you may have mud on the inside of the rim causing a wobble or vibration but may also have bent tie rods which can cause this as well.This will need inspected further.
As for the shifting issue it is very possible the transmission was damaged when stuck or over heated trying to remove it from the mud and may have damaged internal components needing replaced.I have seen clutch/disc failure when trying to get un stuck before to where the transmission needs removed,inspected and repaired.
Unfortunately it is difficult to pinpoint from here for sure which components are at fault here without the vehicle being fully inspected.It is best to have the vehicle put up on a hoist to have the complete under carriage inspected to see what needs repaired or replaced and if everything appears visually o.k. then the transmission will most likely need removed and inspected internally and repaired.
If more help is needed, use the reply tab to continue our conversation. If no further assistance is needed, kindly rate my service.You can rate at any time and we can continue to work on your question as this will not close out your question.Keep in mind in some cases it can be difficult to fully diagnose or help repair your vehicle without seeing it and I work on delivering the best possible answer from what I see to work with.