Emily is always negative about the Algarve in winter. Most of us have had positive experiences.
I don't like Albufeira, so I can't recommend any apartments. It caters for too many hen and stag parties these days. Lagos and Tavira are nicer options IMO. Only the small resorts such as Salema will have a closed for the winter feeling. I have stayed in Cabanas near Tavira for the last two years in March and have had no damp, musty issues and all the restaurants have been open.
Before you rent somewhere, check that it has had several bookings over the winter so that it will have been heated. Take an apartment with heating, as it gets cold at night - there are some great deals for a 4 week rental. I use Homeaway for my Portuguese rentals and have had no problems.
Car hire is incredibly cheap in the winter. Drive up the west coast to Aljezur and round to Monchique and Foia, the highest point on the Algarve. Silves is another castle worth visiting.
If you stay on the eastern Algarve, you could visit Spain for a day on the small ferry that runs over to Ayemonte. Drive to Alcoutim overlooking Spain and the river.
Drive into the Alentejo for a less developed Portugal.
Have a couple of days in Lisbon or Seville.
Walking is good in the Algarve, especially on the cliff tops to the west.