Are you looking for something in Xania? If so you're unlikely to find anywhere offering full board and, broadly speaking, there's a choice of boutique hotels of which there are many, especially around the harbour area. For the last five years we've stayed for 4 or 5 nights at:
Casa del Porto in Sintrivani Square, very close to Remezzo café. A long flight of external stairs to the entrance; excellent view of harbour but must be the only hotel from which the lighthouse can't be seen; comfortable but (back then, at least) not much in the way of facilities although it was the owners' first year.
Casa Leone, Near the Naval Museum,opposite the Mosque; excellent renovation of Venetian merchant's house. Very comfortable; breakfast served in reception area; there are 2-3 rooms at reception level but higher up means climbing a steep spiral staircase; views over the whole harbour. The husband/wife partnership who own the hotel are frequently available, especially around breakfast and they have some fascinating albums of historical pictures of the area and particularly the hotel which they tried to restore as closely as possible to the original building.
Ambassador's Residence, close to the old Custom House and opposite the Fortezza. Very comfortable, spacious rooms in a well renovated interior in tasteful modern style. Access is via lots of stairs or (powered) three stage access for those with mobility difficulties (although signs say for disabled use only in practise anyone can use these facilities in preference to the stairs but they are slow). Very helpful reception staff with good knowledge of local restaurants and other facilities. When we stayed an excellent breakfast was provided at a local café (Pallas) or, I believe they would ferry breakfast to the hotel be we just strolled the 50m to the café. One of the partners who own the place is a descendant of the first German Ambassador to occupy the building.
For the last two years we've stayed at the Porto Veneziano at the eastern end of the harbour (close to the Sailing Club Lounge). This is a more modern hotel and we chose it because stairs are user friendly and there's a lift to all floors. Rooms are well appointed with comfortable beds and there are more guest facilities. There's a large breakfast room, buffet style with a wide choice of breakfast foods. There are also waiting staff who will fetch hot drinks and fresh juices for the lazy amongst us. There's also a café/bar on the ground floor open every day until about midnight; when we've stayed there have been a couple of evenings in the café with a pianist playing "lounge" music. Plentiful friendly, helpful and efficient staff. We booked for this year during check-out last year.
All the above have TV, free w-fi etc. and facilities for making hot drinks.
There are many similar places not only on the harbourside but one of two streets back from the water.
We haven't found anywhere near the harbour that offers full board but there are plenty of restaurants/tavernas open during the Christmas period. Although fewer are open on Christmas Day there's no fear of going hungry but don't expect to find anywhere offering a traditional British Christmas dinner.