Agree with Noddy, I'd never proceed without exclusivity, especially if the offer is at the top of your budget. Do you really want to do business when there's no trust between you? £2k or £3k spent on a fall-through will reduce your budget going forwards.
I think refusing to stop marketing is a combination of trying it on for the best possible price (come on, wouldn't you in their shoes???) and being gutted by the previous fall-through.
Would I be wrong to assume that your DH wears his heart on his sleeve and was a little too enthusiastic during viewings?
If they suspect they can get more out of you they and the agents will try everything especially the notorious "second buyer" scam.
One way round this is to ask for another viewing because you are considering raising your offer; but then "fall out of love" and not give any feedback for a few days. Leaves agent/vendors deflated and worrying that the sale will fall through altogether, at which point you renew your original offer (or, if feeling exceptionally ballsy, a lower one).
BTW, there was a mirror image of this thread just before Christmas with a vendor posting about negotiation deadlock. A search should bring it up and you might find some helpful info.
Buying and selling is hideous in this country - like a protracted poker game. Just hold your nerve.