Car Jerking/Shuddering?

Hey everyone!

Not sure if this is the right category for this, but I was just hoping to get clarification on a few things.

1. It’s been happening more recently, but it only does it when I don’t have my foot on the gas and brake pedals and I’m on ”creeping mode”. Let’s say I’m stuck in really bad traffic or I’m trying to find parking. I’ll accelerate then break then take my foot off both pedals to move slowly. My car will shudder a little and the RPM needle will move up and down. It also sometimes jerk forward like something is pushing my car from the back.

2. My car will shake when changing gears and the RPM needle will drop a little. Doesn‘t happen a lot but will sometimes.

3. After filling up and driving away from the gas station and to a stop light, the RPM needle will move down then up after like the car is breathing and will only go away when I drive away and doesn’t happen again until I fill up again.

Recently got an oil change and had the mechanic look at the reason why these things are happening to me. For the third point, they said it could be my compressor or a multitude of things. I stated I don’t think its my AC bc its never on when getting gas. As for the first point, they said I would need to replace my transmission mounts. This could be possible for the first two points. I got into an accident about a year and half ago where three cars (including me) hit something on the street. I’ve also hit “hidden“ bumps, not at full speed. Not sure if those can affect my transmission mounts. I’m not sure if replacing the transmission mounts can fix all of my issues, but I’m hoping it will for at least 2/3. The quote is expensive and I do trust this car shop since they did my oil change and fixed my car AC, but I don’t want to get it fixed and none of my issues are fixed.

Here are a few things about my car:

2013 Hyundai Veloster Turbo; about 82k miles
Last oil change was May 2023
Replaced the spark plugs in January 2023
Replaced MAF sensor and gas cap in November 2022
Crankshaft sensor in June 2022

Thank you! 🚗