Read your original post and from where you were at three months ago to the wins you've had with her today is really great progress!
First - it is very likely to improve, but you still have to work with her on her socialization skills. I rescue cats and have socialized a number of feral cats and she sounds like what we'd classify as "under socialized." I've used a method called "Socialization Saves Lives" for socializing feral cats and I highly suggest it! She seems to be somewhere between a Phase 2-3 so you don't necessarily need to start from the beginning with her.
There are strategies and "sessions" you can have with her that will work on all these skills, which the linked guide lays out but I can summarize a bit below. It seems like the biggest ones from your post are:
- Approaching
- Picking up
- Getting her into a carrier
Getting her into a carrier seems like the most pressing issue since she needs to be examined by a vet. You don't necessarily need to pick her up for this. Keep her carrier out in a room she likes at all times. Feed her her meals around it and eventually start feeding her her meals inside it. You can even put a blanket in there so it becomes a bed. Basically make her associate her carrier with positive things and make it so that she's used to going in and out of it. Schedule a vet exam and maybe a half hour or so before you have to leave for the appt give her a meal in the carrier and she'll think it's just a normal day, except you close the door behind her.
For approaching, the Socialization Saves Lives guide above has a whole video description of tricks to try but in essence its this - find a treat she likes (could be anything from kibble, to Churu/lickable treats, to even chicken/ham baby food (so long as there are no other ingredients besides the meat/water/cornstarch.) When you have to walk by her or approach her, do it slowly and with your side facing her. You can slowly crouch to offer her a treat as you do so. Switch it up when you do sessions like this with sometimes you just offering her a treat and not interacting at all and other times you offering the treat as you approach and then doing whatever level of petting she typically allows. It could take a few days to a few weeks before she's not always running away, but just keep at it!
For carrying this is a very advanced "socialization" move so work on it last, once she's comfortable with the other elements. Get her used to coming into your lap (the "lap work" sections of the guide linked). Once she is then get her used to you putting an arm around and under her belly. Don't fully lift, but just get her used to the presence of your hand there. Over the course of many sessions put more and more upward pressure, cradling her more towards your chest. Then you can work on getting your hand under her bottom. Essentially, getting her used to being lifted and brought towards your chest while you're seated and on the ground will make lifting her from a standing position easier.
Basically it's just staying the course and trying to work with her on these specific skills daily using positive associations and slow, progressive movements!