Hi All, OP here. Thanks for all your replies. I left town yesterday in the early morning, and the pet sitter was supposed to come by twice yesterday, AM and PM. She is not staying overnight; she's coming twice a day for an hour or so at a time (she said she might stay longer) to feed my cat, play with her, and administer medicine.
I called yesterday evening, and she said that everything had gone just fine. My cat let her give her the medicine, and even accepted a few pats (my cat takes some time to warm up to people). So, that was a huge relief. The sitter didn't mention anything about the computer at all (I had taken it with me, with a note that I needed to finish some work on my laptop while I was away).
A few posters mentioned that there are probably other pet sitters available in my area. That is correct. What happened was that I first called this sitter a couple of months ago, after seeing her website and reading all her positive references. She's not with petsitters.org, but she is with Pet Sitters International, and bonded/insured. Anyway, the phone conversation went well, so I agreed to hire her. She didn't want to set up an in-person consultation until just a few days before my trip, because she wanted my cat to be able to remember her when she came to pet-sit. So, I didn't actually get to have a long in-person chat with her, and hear her request for internet/computer access, until less than 48 hours before my actual trip. That's why I said that I felt "stuck" with her, because the other pet sitting services in the area require that reservations be made more in advance than that, especially for first-time customers.
Anyway, I'm just relieved that my cat seems to be doing well!
Edited to add: Reading this over, I realize that I really should have interviewed more than one pet sitter back when I was first looking around! I just felt satisfied with our phone conversation and kind of stopped my search at that point. A big mistake on my part-- I should have done some comparison shopping!
Edited a second time to add: Just to make the timeline clear (I said in my original message that I was leaving "next week"), so that no shenanigans are called
: Called the pet sitter originally in late February. Met with the cat sitter in person Wednesday (5/21) at 1pm; left for my trip Saturday (5/24) at 8am.