As Ken stated, the area around the train station in Venice is not considered close to main sights. While Venice is pretty compact, I would say that the areas near or between the Rialto bridge and St. Mark's Square would probably be the most centrally located for most activities. In Venice, we stayed at Locanda Orseolo (the #1 rated B&B on Trip Advisor and for good reason). From the B&B, it's a 5-minute walk to St. Mark's Square and 10 minutes to the Rialto area. The B&B itself was wonderful and we really enjoyed staying there. They have a water taxi landing and we arrived and departed via private water taxi which was a treat; we appreciated not having to schlep our luggage over bridges, etc. The only downside is that the B&B doesn't have an elevator so we had to walk up several flights of stairs to get to our room on the 3rd floor. Since you have trouble walking, you could ask for a room on a lower floor to minimize the number of stairs. The B&B definitely has the local charm/old world decor that you spoke about.
In Sorrento, we stayed at Palazzo Montefusco and again loved it! It's about a 10-minute walk from the train station to the hotel but the hotel will send a porter to meet you at the train and carry your luggage back to the hotel (this was a really nice unexpected touch!). The hotel is located just off of Piazza Tasso and close to everything. The hotel straddles two streets, one of which is pedestrian only all the time and the other street is pedestrian only at night. We could walk outside and be in the midst of shops, restaurants, gelaterias, etc. It was a 10 or 15-minute walk down to Marina Grande. The décor was more on the modern side but extremely nice and comfortable.
In Rome, we stayed at Albergo del Senato right next to the Pantheon. We literally felt that we walk reach out and touch the Pantheon from our balcony! It's probably on the pricier side but was a splurge and we really enjoyed it. The location couldn't be beat and I would definitely stay there again. The décor is "old world" but very comfortable.
By the way, since you have trouble doing a lot of walking, have you considered hiring a private guide/driver for the Amalfi Coast and/or Pompeii & Herculaneum? I have some health issues and decided that having a private driver was the best option for me because it would minimize all the walking and by hiring a private driver/guide, we definitely saw way more than we would have on our own. We hired a wonderful guide for Pompeii and Herculaneum and the driver picked us up at our hotel in Sorrento, drove us to both sites and also took us to a wonderful winery at the base of Vesuvius for lunch which we would not have had time for if we were using public transportation. We visited Herculaneum in the morning, then had a nice leisurely lunch and visited Pompeii in the afternoon and then were delivered back to our hotel in Sorrento at the end of the day. It was a tiring day but a lot less tiring than it would have been if we had done it all on our own! We also used a private driver for the Amalfi Coast and it was so enjoyable getting to sit in an air conditioned Mercedes van and be delivered to every stop and not having to deal with the crammed and hot bus. I can give suggestions if you want.
