hotel suggestions York and Bath

That budget is a little low. The Guy Fawkes Inn (York) has rooms for your dates at £152. Have you looked at the Premier Inn chain?

Have a look on Booking.com.
For those dates in York, for instance I've just seen Middletons for £215 for 2 nights, or The Bar Convent [was a Convent, not now!!] for £188, both of which look very characterful properties. The Bar Convent is only just round the corner from the Station.
But there are other choices in your budget, including one of the PI's.
Both seem to have good Trip Advisor feedback.
Does anyone have feedback on those two?

I've stayed at The Bar Convent. In October 2015. As noted, it's just a walk down the road from the train station. It's also across the road from the Micklegate, your entry into the old walled city.

I liked it. So much that I'm staying there again this year. Their rooms are mostly ensuite, but they do have some with shared bathrooms. I had a single ensuite with a recently remodeled very nice bathroom. The cooked breakfast was good, the staff were kind and accommodating, and I'm looking forward to my next stay. (For my stay in late May, I reserved an ensuite single for £85/night.)

If not there, I hope you find a good place for your visit in June.

BTW I know you won't have a car, but when I visit the Bar Convent website, I always savor the information that the nearest car park is on Nunnery Lane. :-) (Yeah, I'm shallow.)

Bath, the Brooks Guesthouse that is walking distance from downtown and near the Scallop Shell Restaurant.
York, the Minster Hotel, they have spacious rooms and parking for your car. Also, very close to the city center.

Thank you for the suggestions. I just booked the Bar Convent and am looking at the options in Bath.

We are staying at the Brontë guesthouse on York in less than two weeks. Trip report may be too late for you to make a reservation, but I’ll post anyway.

The Bar Convent is great and a very different experience. The cafe is popular with locals but hasn't yet been discovered by tourists. It has a very interesting history. Do take time to go round the exhibitiion and don't miss the beautiful chapel upstairs!

Don't flog me for asking about a chain hotel instead of a B&B, but does anyone have feedback on the Hampton in Bath? I have Hilton member points I could use. Corner of James and Avon, near Bath College. It looks convenient to the bus and train stations, and the baths and abbey.

I've not stayed there but it does get consistently good reviews on Trip Advisor - and not by just one posters either...

In Bath, we like the White Guest House (https://www.whiteguesthouse.co.uk/).

We loved St. Mary's Guest House in York. It is a short walk from all the sights and yet on a lovely quiet street. The host is super and the breakfast ... well you won't be hungry for a long while.

Thank you for all the suggestions I booked the Bar Convent in York and the YMCA (private en suite room) both under 100 pounds a night.

Many of other places suggested looked great. Can't wait to visit both places.

Galtres Lodge Hotel is a great hotel in York.