One suitcase in the hold and 4 sets of hand luggage for 5 days is definitely doable! Especially as you can just share one large set of toiletries (eg no need to pack half a dozen bottles of shampoo when one will be more than enough for the four of you for a week) that can go in the hold bag, 5 sets of clothes each (wearing your bulkiest things on the plane) and then anything extra can go in hand luggage. You're only going for a week, you don't need to take your entire wardrobes.
I have just come back from 3 weeks in Spain taking only one hand luggage case and a handbag, it is entirely doable. I was working as a director at a summer camp so I couldn't swan around in swimsuits and flipflops the entire time but had to be casual enough for the day to day activities and also be "smart" enough for arrival/departure day when all the parents were around. I managed to get 3 pairs of jeggings/leggings, 3 pairs of shorts, 2 skirts, 10 t-shirts/vest tops, a couple of swimsuits, a couple of light cardigans/cover ups, beach towel, two weeks worth of underwear & socks, a pair of flip flops and a pair of sandals into my case. Then used my handbag for my laptop, tablet, phone (and all the cables), toiletry bag (with the bare minimum of make up in because I use Boots click and collect to order full sized products to the airside store in the airport, so I save on space in my case and it actually works out cheaper if they have offers on), purse, phone, keys, passport, headphones and other bits and bobs. I wore my jeans, walking shoes, lightweight jacket and hoodie on the plane
I flew with Easyjet that have no weight limit on hand luggage, as long as it fits, and paid a little extra for up front seating so I could get the second hand luggage piece. The airport were doing a "hands free service" where you could check your hand luggage in for something like £4 (rather than having to lug it round the airport and potentially gate check it last minute) so I checked my case which ended up weighing only 13kg and just kept my handbag with me.