That was pretty much my experience for the first eight months or so on T. I'd be sore, the muscle would just feel tense, and I'd feel very limited. It was not uncommon for me to have a slight limp the day after my shot. The solution I stumbled upon was exercise. I was terribly lazy the first eight months I was on T and wasn't doing much, if anything, in the form of exercise. When I decided to have bottom surgery and wanted to lose weight, I started alternating cardio and lifting every day.
If I go for a run on the day of my shot and go to the gym and do legs the day after, I have no pain whatsoever - not in the muscle or on the surface level. I'm not sure what it is about that routine that helps, but I could never go back to being sore for up to 3 days after the fact if it means I just have to run a few miles and lift a few heavy things.
As far as technique for shots goes - I alternate legs, I pinch/gather about an inch of skin instead of flattening it like some people do, I do not aspirate, and I usually end up injecting somewhere between 5-8" up from the knee, if that helps.