I cannot offer any advice on hotels in Venice that are along the grand canal, but I do have a few hotels I would recommend for Rome. You absolutely can find rooms under $250 they just may not be in a super fancy hotel. I’m more of a budget traveler and I usually lean toward smaller boutique hotels that offer the amenities I need.
I’ve really enjoyed the places I’ve stayed but definitely recommend looking at reviews. tripadvisor has always been my go to for the research and then I book direct with the hotels, in some cases getting free nights by doing so.
Here are a few, all appear to be under the $250/night price range:
Residenza zanardelli - right around the corner from Piazza Navona. Stayed here in 2007 and would stay again without hesitation. Although our cab from the airport dropped us right out front without issue, it can be easy to miss/hard to find as there was not a huge sign on the building: we had to look for the name on the building by the call bells. The building has a small lift (elevator) and our stay included breakfast
https://www.tripadvisor.com/Hotel_Review-g187791-d239503-Reviews-Residenza_Zanardelli-Rome_Lazio.html
Hotel Navona - in a quiet area between the Pantheon and Piazza Navona. Stayed here in 2015 and would stay again without hesitation. The building has a lift and our stay included breakfast. Originally we were placed in a room that was super cute but quite quirky (a very low archway-maybe 5ft 5 inches high-separating the room from bathroom with a curtain instead of a door. No shower but a bathtub) and we asked to be moved to a different room, which they did as soon as they were able.
https://www.tripadvisor.com/Hotel_Review-g187791-d198738-Reviews-Hotel_Navona-Rome_Lazio.html
Imperium suites Navona - in a quiet area a few blocks from Piazza Navona. Stayed here in 2014 and would stay again without hesitation. Initially it may be difficult to find on your own but our cab found it with ease. I do not remember the building having a lift but we only had to go up one flight of stairs - although my understanding is they also have rooms in an annex across the small street so I am unsure on that building. Our stay included breakfast and it was by far my favorite as they left a menu the day before to choose our items and they delivered them to our room at the time of our choosing the next morning. Our bed was very hard here but we loved the location.
https://www.tripadvisor.com/Hotel_Review-g187791-d1109342-Reviews-Imperium_Suite_Navona-Rome_Lazio.html
Starhotels Michelangelo - near the Vatican. Stayed here in 2016 and while it was further from the area we normally stay - and a larger hotel - the rooms were quite charming. We requested a room with a balcony, from which we were able to enjoy a partial view of the St Peters dome, it was magical to view at night from our balcony. Here they offered room service as well as breakfast in a large banquet room. Because of the location we took a cab a few times over the course of our stay if we needed to go a long distance but in most cases we were able to navigate the city by foot and broke up the walk if it was long with a stop for gelato or cappuccino.
https://www.tripadvisor.com/Hotel_Review-g187791-d203056-Reviews-Starhotels_Michelangelo-Rome_Lazio.html
All of these hotels had AC and the rooms were larger than anywhere else we’ve stayed in Italy.