Minehead is dog friendly from golf course on,it's only the area on front that isn't.Blue Anchor and Dunster beaches just a bit further on towards Taunton are both dog friendly,Dunster they can be off-lead below high tide mark only,Blue Anchor all the way.There's also a fab dog friendly Cafe at Blue Anchor - Driftwood Cafe,great food and free parking too,pay at Dunster Beach.You can walk from Minehead to here and catch the steam train back as well.
Just seen you're in Dulverton (gorgeous isn't it ) Do try to visit Dunster village if coming to beach,it really is lovely.
Kilve beach is also a good place to visit,if you park at East Quantoxhead church it's just 10p and you can walk along beach to Kilve then along clifftops as far as you like,Kilve is interesting for the strange rock formations.
Head to North Devon too,Saunton Sands is an amazing dog friendly beach,car park fills pretty quick though so get there early,well worth the drive.