I live in Hastings and I love it. We moved from Brighton shortly after DC1 was born in order to reduce our monthly outgoings (which it did).
It is true, there are some pretty grotty bits, but there are in any town aren't there?
The Old Town is lovely, although you'd be lucky to get a direct sea view from there. I'd recommend looking out for places on the West Hill, so you'd be on the cliff directly above the Old Town. There is a stretch of seafront between the Marina and Grovesnor Gardens in St Leonards, where there are some beautiful houses, a lot of which have been done up in the last few years.
Warrior Square in St Leonards has some great properties (you're bound to come across some), sea views, all grand and Gerogian, but has a reputation for being a bit rough (which the estate agent wont tell you) as it is near to London Road- which is imo an absolute dump.
If you're at all flexible about the sea view- there are some absolutely gorgeous places overlooking Alexandra Park.
You'll probably be best avoiding most of Ore and Hollington. I reckon get a map asap, as it is a small town, estate agents will often tell you that a property is in a nice area when it is just near it. eg. for 'Near the West Hill', read 'Ore'.
Good luck with your search, and don't be put off by the fact that people love to slag off Hastings