Odd Noise and Vibration Upon Starting in First Gear

When I start the car in first gear, there's this noticeable noise that gives the impression of something being dragged along. It's quite distinct. However, as soon as I accelerate, the noise seems to fade away gradually. Alongside this noise, I can also feel vibrations transmitting through the pedals, adding to the overall sensation. It appears that the source of this disturbance is situated in the lower front section of the car. Interestingly, this issue becomes more pronounced when driving uphill, as if the incline exacerbates the problem.

I should note that I've recently replaced the lower and upper right engine mounts, so they're new. The only one I'm unsure about is the transmission mount, but it doesn't seem to be excessively loose.