Even if she sold her own house, while having an offer on the new apartment - the chances of both sales going through on the one day or the one week is fairly slim, in my opinion.
So many sales fall through, people get cold feet, she may change her mind herself on the apartment etc.
Having to suddenly be out of your own house, indeed makes you need to decide on buying something pretty quick, which makes people rush into decisions. And puts a huge amount of importance dependent on the timing of the sale, which causes huge stress.
She may have to face renting in between if she needs to.
I know it'd be a big stressor for a 74yr old lady used to her own house. Is there a relative she could stay with in between?
Using a storage facility to store the extra clutter and furniture while she is selling is a good idea. Then when she is moving to the rental place short term, get a furnished place and she can just put all her own furniture and house contents in the storage container.
The move to the rental place or the relative's house is not so bad then, as it's not a complete house-contents move.