Hope I'm not too late, try Powderham Castle on the other side of the estuary. Drive past Darts Farm & keep going through Topsham, then turn left. The castle is great, really, 'castle''ish.... & they have ponies, chickens, lovely walks & a big wooden play castle thing,all on the banks of the Exe - we've had some brilliant times there.
Dawlish is a bit further on from there & is nice old sea sidey place - very 'Agatha Christie & cream scones' iykwim. Love it lots...
Also, there is the outdoor play park place on the front at Exmouth, quite a way along the front, next to the crazy golf & the cricket club. You pay a couple of quid entry & it is all grassed so you can take a book & a picnic.
Last one; there is a private club, the Cranford I think it's called, have a look on line, they have a lovely outdoor pool, good bar & food, tennis courts, blah, blah. Not sure, they may do day or week memberships, but definately worth a look - it's lovely.
Have a great time - wish we still had friends to house sit for in Exmouth. Met a fabulous MN'er down there, did me a huge favour at the time.