Is the housing market seriously on the up round you? (Genuine question, as all the EAs will be talking it up for all they're worth, but what do the sold prices say?)
I think I might start making noises to the EA about BoE's change of heart, possibility of interest rate rises, house prices coming down, might not be a great time to buy a house, maybe you're better off not buying at the moment but renting instead - £1k a month on rental but £10k a month coming off house prices, etc. They'll bluster back, but if they think that they're losing good buyers (ie buyers with a house already STC), they'll start acting on that. Better a bird in the hand, and it's amazing how many people make offers without the financing in place and how many accepted offers lead to nothing. Whatever the EA believes or hopes, it would be in their interests to start mentioning to the vendors that there is no security that they would achieve the same offer if you dropped out...
OTOH I would also dangle the long completion carrot. That is what we did, but other way round. Our vendors were pissing about big time, we had buyers who had buyers who had buyers who had buyers... and the time was ticking on the mortgage offers from the bottom up. We agreed an exchange date with everyone below and a long completion date in the school hols, which meant we would take the risk of being homeless (but with a lot of cash in our pockets, parents to stay with over the summer if necessary and time enough to find somewhere to rent) but we would guarantee everyone below would get their mortgages and be able to buy. Amazingly, once they realised that we had indeed done what we'd said we'd do, and cut them out of the chain completely, the vendors got their act together, our solicitor was suddenly deluged with the documents she'd asked for five months previously and a few days later we exchanged, with completion on the same day as completion of our sale (one of the conditions of exchange, or alternative of vendor paying storage costs until their preferred completion date).
The problem is that unless you sell, you can't buy anything (and your vendors are in the same position). You can't risk losing your buyer. Worst case scenario is that your buyer gives up, you don't sell, you can't buy the house, you've lost all the legal etc costs so far, which you'd have to pay again if you tried selling again, with no guarantee of finding another buyer quickly or indeed another chain going through. Weigh that against 'worst' case of selling your house, having the cash, being in a strong position, but paying £1k per month rent and breathing space, and renting may be worth considering.
Don't give up hope, but be prepared to act tough. You may think the EAs are 'nice' but they are not acting for you. They are primarily acting for themselves and will follow the easiest course of action to achieve the sale of their client's property. At the moment, that is going along with the vendors' faffing but you could easily make chivvying things along their preferred method 