Recommendations for a central hotel/BB in Edinburgh near Waverley Station

I liked my stay at Cheval Old Town Chambers. It was right on High Street across from St. Giles. It may be higher than your price range but I remember booking a special which was a discounted price in your range. They have kitchens and washer/dryers which helped me save some money. Location can't be beat.

The Parliament House Hotel may fit what you are looking for. It may be a bit over budget but you can check it out.
Howie’s restaurant next door is a good choice for dinner.
Short walk to both Waverly and Royal Mile.

I liked my stay at Cheval Old Town Chambers. It was right on High
Street across from St. Giles.

You refer to Old Town Chambers, Autograph Collection? I cannot find Cheval Old Town Chambers online.

The Parliament House Hotel may fit what you are looking for.

Yes but a bit overpriced. :(
Weekedays yet a bit pricey.

You may like Market Street Hotel. A little over your price but across from Waverly station. We stayed in a cozy room and it was plenty roomy. We asked for a quite room and it was very quiet.

The Marriott Courtyard is quite nice, and it's also very convenient to both Old Town and New Town. Extremely close to Waverley Station.

Edinburgh can be pretty expensive. If you’re happy with comfortable chain hotels that aren’t particularly characterful, there are a couple of well-located Motel One hotels and ditto Premier Inns.

The Motel Ones are Edinburgh - Princes and Edinburgh - Royal.

The ibis Styles hotel at St. Andrew's Square is very nice and reasonably priced. There is a recent review of it in this trip report from Pam: https://community.ricksteves.com/travel-forum/trip-reports/trip-report-seymour-travels-orkney-and-shetland-plus-edinburgh

It's about a 6 minute walk from Waverley and the location is wonderful. I love St. Andrew's Square as it has a nice residential feel to it, but there are plenty of restaurants and cafes in the area.

If you want something cheaper, there is the Premier Inn at York Place (a 10-11 minute walk from Waverley), which is very clean and very comfortable. I've stayed there before and enjoyed my stay. It's part of a chain but I've stayed at a number of them, and they are always consistently good.

I stayed at Hotel Ibis - Hunter Square. It’s just off the Mile and directly over the bridge from Waverly. It was centrally located and didn’t break the bank. It may exceed your budget a little, but breakfast is included and if you join their hotel chain list (free), you get a discount.

I’ve been reading reviews of the Parliament House Hotel to see if there are mentions of elevator issues. When I stayed there last week for 2 nights the elevator was “out”. I see one review on TA mentioning that the elevator has been out for a while and another review specifically mentioning the elevator does not even go to the floor where my room was located. I frankly did not even notice that but in thinking about the landing area there was a solid rail around the center of the stairwell which is the elevator shaft. The # steps were: ground floor to 1 = 25 steps, 1-2=25 steps, 2-3=23 steps.

Pricewise it may meet your budget depending on day of the week.

The access to the front door is way easier from Waterloo Place and looping around the corner from Howie’s restaurant.