"Is it strange to ask for a few weeks "grace" from the buyer of my home in order to stay there whilst the other transaction goes through?"
Yes, very. When the buyer pays over the closing funds, they are entitled to Vacant Possession.
"Is it possible to move into a new home whilst the sale is still going through (and at an advanced stage of course)?"
As above, you only get possession when you've paid over the closing funds.
It is possible to reach agreement with either your purchaser or your vendor to allow either scenario but , as a practising solicitor, I would advise clients against any deviation from the norm. We all have horror stories of agreements turning badly wrong when something unforeseen crops up.
mf