Can someone recommend an area in Amsterdam & Brugge that we can just stroll along the streets and find a room for rent?
Many hotels around the 1st canal belt will have rooms to rent on spot, especially some close to Centraal Station. problem 1: they will be woefully overpriced for the quality they offer problem 2: some of them are shoddy problem 3: it is hard to find a good room on spot, unless you are willing to walk like 2 hours or more coming and going out of hotels I'd really not recommend that in Amsterdam.
I have the impression Lynda is thinking more of what might be called a rooming or boarding house - a private residence that lets one or two rooms to visitors. I was just in both Brugge and Amsterdam. I rode a bike or walked around much of Brugge, and while I saw signs on buildings for rent or sale, I didn't see any signs for rooms to let to tourists. However, there could be some in residential sections a bit further than I ventured. I also get the feeling that you want to wing it, but that may not be a good idea. There is a website called Roomorama through which private rooms are on offer nightly or weekly. The link below is to do a search in Brugge. Change the city to search in Amsterdam; the Jordaan district is really nice. https://www.roomorama.com/pages/private-rooms-in-bruges Good luck!