I moved my late mother’s 2 cats 3 years ago to here with the intention of rehoming them elsewhere. They were both very timid, very much like you describe your Amber. When they arrived in their carry baskets I took them up to a spare bedroom & left them in the baskets for an hour or so to settle after the 2 hour car journey, then I put out their litter tray & food (luckily I had an ensuite for them to use, they stayed in there for a couple of weeks to really settle(bear in mind they didn’t know me & I had never been able to get near Lilly at my mother’s. I obviously visited them every day in there, I then started giving them access to the parts of the house I didn’t mind them in for short periods. After 4 weeks, Lilly escaped from an open window!! Thanks hubby. I eventually found her when she crossed the road in front of my car a few roads away,7 weeks later & not looking very well! I nursed her back to health 🙄. Ollie the other one started to go outside after 5/6 weeks & they now both sleep in the utility room, get fed in there, I’ve managed to do away with any litter trays, which my mother never managed to do (my garden suffers for it as do my planters, no dig doesn’t seem to have made an impression on them 😂). Lilly now spends lots of time outside which she hardly ever did at mums, preferring to be on her bed day & night.
They have come along way in 3 years, they’d be better off in a home where people want them but I could never send them off to a rehoming place.
Long way of saying I’m sure they will all adapt to your new home if you just take it easy, you might need to consider sedating Amber for a few days before & after the move just to make it less traumatic for her& keep her in so she doesn’t go hidey before you move.