2 Hotels or 1?

With 5 nights
Half day arrival Night 1 - Depends on where you stay. Either Andrassy ut day or the Danube Corso Day
Full Day Night 2 - First half of Pest sights
Full Day night 3 - Buda and a trip to Szentendre (depends on time of year).
Full day night 4 - Second Half of Pest sights
Full day Night 5 - Usually a specialty day. Could be museums or Jewish Budapest or Cold War Budapest or some special topic like architecture tour or winery tour or distillery tour or ...... Easy to fill a day this way.

Depart

If you just want a change of scenery, then sure, change hotels, but it isnt necessary.

where do you recommend they are in relation to walking to sites?

I like Andrassy ut and the link below is a good collection of the hotels on or near Andrassy ut. But you arent going to walk to all of the sights. You will walk to the sights close to your hotel on Andrassy ut. The ones further up Andrassy ut you may take the M1 Metro up and walk back. For the Jewish quarter and the sights along the river you will take a tram or two. For Buda mabe a bus or a tram and a walk and a funicular. You will do all of this using a 15 day Budapest pass that cost you about $14. Unless you are 65 or older in which case its all free. The M1 metro is the oldest on the continent of Europe and sort of low key and fun. the trams are definately the best tour busses in the world. Large windows, antique feel, and they go within a 100m of just about every attraction in Pest.

Getting around Budapest to all of the sights is such that one hotel, if in a fairly decent location will put you within 15 minutes of 90%. Any of these are well located: https://community.ricksteves.com/travel-forum/hungary/rs-forum-favorite-hotels-and-their-cost.

And you want to be in Pest, not in Buda unless you have some really, really good unique reason to stay in Buda.

Thank you so much for this info! So helpful!

And I sent you a PM