Is it safe to drive? Probably not. Continuing to drive with a jerking transmission could cause further damage, potentially leading to a complete failure and expensive repairs. The damage could range from worn clutches and bands to more severe internal problems.
What you should do:
* Get a second opinion: Take your vehicle to a reputable transmission specialist (not just a general mechanic) for a proper diagnosis. Explain the symptoms in detail.
* Don't ignore the problem: Delaying repairs will almost certainly worsen the situation and increase the cost of fixing it.
* Drive cautiously: Minimize shifting as much as possible while you get it checked. Avoid aggressive acceleration or hard braking.
The cost of a transmission service varies greatly depending on the vehicle and the extent of the work needed. A simple fluid and filter change is relatively inexpensive, but a more significant repair could cost thousands. Getting a proper diagnosis from a specialist is the first and most important step.