Car is swerving after I change gears and also after accelerating

It's time for an inspection at a tire and alignment shop. The first things to look at are the steering and suspension systems. In particular, worn ball joints, tie rod ends, control arm bushings, and struts can cause what you described. Those allow alignment angles to change between accelerating and braking / coasting.

If nothing is found with those systems, also look for a collapsed engine mount. One of the end mounts can let that end sag which changes the carefully-designed angles of the half-shafts. Uneven angles can cause "torque steer", which means the car will steer to one side depending on the acceleration and braking forces acting on the tires.