If you're not keen on AirBnB, then perhaps have a look on Booking.com, which has a good range of lower priced accommodation. UK style BnBs are less common than things called pensão, residencial or quartos. Of course, there may also be hostels with private rooms.
You could also use the Hotel function on Google Maps to see what else is there, and contact directly (an email address or website is often lower down in the Google Maps inform for places). In the Algarve, I'd expect most places to be able to communicate in English.
Bear in mind that a lot of accommodation is not currently taking bookings due to the COVID situation, so this may account for apparent lack of choice or availability. Places showing as unavailable on Booking etc may be open to a longer-than-a-couple-of-nights booking if approached directly.