Hi there, and welcome to Talk Budgies! You did a great service to Luna by bringing her to a loving home and getting her vet care!
It sounds like she is doing much better (chirping, alert, etc.). It's really hard to give antibiotics, but it's worth it in the end (I give them to my Sammy so often as he's a sickly bird).
There are a couple more things that you can do to make sure that Luna is thriving:
You can measure her with a kitchen scale every day (put a tupperware on top, tare the scale, and put her inside the tupperware with the lid gently propped on top). This will tell you if she's gaining weight, which is a great sign.
You can also put on some budgie sounds on YouTube for her, which she will probably love. If you can see her singing and calling to them, it's a great sign, as a sickly budgie wouldn't take much interest.
Sounds like the fact that she's eating veggies means that she's on a great road to recovery! Incorporating probiotics is a great step as well as giving her mineral blocks and cuttlebones.
If you need help keeping her warm, I would suggest a hot water bottle on half of the cage. I actually use a small heating pad clipper to half of Sammy's cage year round (at night during the winter and any time he's sick). I have one that only turns on when the temperature of the pad drops so that it isn't on all the time.
Best of luck with little Luna!