Thank you to everyone who had helpful suggestions, I got some antihistamines and sore throat spray and it’s definitely helping, I still feel pretty rough but it’s drained a lot of the congestion. Not looking for validation to leave—I’m not a quitter like that, lol.
To clarify about my history with cat allergies, it has NEVER been this serious. We’ve had 10 cat sits and I’ve lived with cats before, including indoor/outdoor cats. With the indoor/outdoor cats I had to wash my hands after petting them and not stick my finger in my eye. With the indoor-only cats, I literally pick them up and put my face in their fur with no issue. My problem, as far as I can tell, seems to be the closed-air-circulation environment in combo with an indoor/outdoor cat and possibly the construction dust. I seem to be able to tolerate dander just fine in other circumstances, but this particular house is too much.
Obviously don’t accept a cat sit if you have major cat allergies. I’ll probably be more direct in the future about asking homeowners how often they vacuum/clean and ask to see the home over FaceTime before accepting, as well as clarifying how much time the cat spends outside. And for now, load up on antihistamines, keep away from the cat (that’ll be difficult, she’s a darling 
), lock all the animals out of the bedroom, and just suck it up for the next few days…