Good luck! Although it can often be cheaper to book direct, I think in your case you'd be better off with a travel agent's advice - they know all the hotels with family rooms and all the self-catering places with bigger apartments
We're off to the Caleta Dorada in Fuerteventura next Easter with First Choice - we've booked a 2-bedroomed apartment with twin beds in one bedroom, kids bunk-beds (with TV, Playstation and DVD player) in the other bedroom; and sofa-beds in the lounge area. Actually there's only 3 of us (me and two teenagers) but we're greedy with space and I got a good deal (and I wanted to sleep in a proper bed in my own room instead of a lumpy sofa-bed) We've booked All-Inclusive and paid adult price for all three of us, but they do self-catering there too
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There's a few other places in the First Choice brochure with larger 'family rooms', so it might be worth you having a look...