OP, you may also want to look at Lanzarote for more choice and if you avoid Costa Teguise or Puerto del Carmen, you will also find it 'very relaxed' - Playa Blanca is a nice resort with a good choice of hotels and apartments and you can see Fuerteventura across the sea and go there on a ferry day trip.
Lanzarote has a bit of a reputation for being windy, but then so does Fuerteventura, and probably more so, given that the place name is Spanish for fierce wind.
We do self catering or B&B and it's swings and roundabouts whether a package or independent booking is cheaper but there's usually only about £100 or so in it.
We're actually going away on a jet2holidays package later today, which cost £820 for 2 of us for flights, transfer, hotel and luggage.
The same flights alone with luggage, which we need as we're taking scuba gear would have been over £500 and the transfer another £100 as it involves 2 taxis and a ferry which leaves about £200 for a week's B&B in a nice looking boutique hotel.
I couldn't have booked everything separately for that price unless I got a basic self catering apartment and different flights that involved a 3 am start in at least one direction - a lot of Jet2 flights are at a civilised hour, which is why they can cost more than the 6 am Ryanair or Easyjet options.