My estate agents opinion seems to be you sell and then move out even if you have no where to go
Your estate agent is being a duplicitous arse.
It is in THEIR interest for you to do this as then they get their commission.
It is NOT in your interest to do this.
Please don't be talked into having a deadline, to be nice to your buyer. You are being taken for a mug if you do this.
You must act in your family's interest, not allow people to screw them over because you're being too nice.
My friend's mum got talked into going into rented. By the time she found somewhere, she could no longer afford to live in the nice inner London borough she'd live in for 40 years and planned to retire into, and had to move somewhere rougher and further out as she coudln't affrord it - because prices went up while she was renting. This is what you're risking.
Let the buyers know that you are doing everything you can. And be prepared to let them go, you will find more buyers quickly in the kind of market you describe. You owe these people nothing! It's not your fault the market is as it is, and you sound like you're doing your utmost to find a place. Let them go.
And, take everyting your agent says with a pinch of salt from now on. It's very clear they're acting in their own interests, not yours.