Yep! Covering the cage would probably help; remember that budgies are naturally desert birds, but you still do need to be careful about heat. If you see her panting, be sure to cool her down however you can; shade, a fan, water, whatever.
Although many breeders tell you to quarantine budgies for up to 6 months, I usually tell people a few weeks. However, with the shady pet stores and the threat of mites you were telling me about, I'd be very concerned about your new bird infecting your current bird with something nasty. If you aren't willing to keep a new bird in prolonged quarantine under your current circumstances, I would not get another one at all. Your baby girl has come so far. It would be awful if something bad happened to her again.