Signing the contract is usually done in advance so that a telephone exchange can be done by the solicitors once everything is in place.
Sounds as though you using the funds from the sale of one house plus a mortgage for the purchase of the other?
That is effectively a chain and your solicitor is probably looking to exchange both on the same day. It sounds as though you are confusing things a bit though - your solicitor with request the draw down of the mortgage monies in advance to ensure exchange can go ahead as you are then committed. When have they said you will complete?
You will be advised when exchange has happened as you will need to insure the place you are buying.
The reason the buyer is doing this to make sure they don't end up.in breach of contract. If they exchange today and then the mortgage money doesn't turn up they can't move in but will be liable for costs.
If they don't exchange then if the money doesn't turn up they are.not in breach.
Mortgage company draw down times vary and it might be they are trying to do it in less time so need the confirmation. They aren't being unreasonable. It's frustrating for you but standard practice.