Elinor met John at Gray's Jewellers in Sackville street to carry on a negotiation to sell 'a few old-fashioned jewels of her mother's.'

Probably Mrs Dashwood would've liked to have left her jewellery to her daughters. To speculate, Elinor has advised her mother it would be prudent to sell the jewllery.
I imagine a London jewller such as Gray's could give the best price, though still not the full price of what the stones are worth.
The story doesn't mention the Dashwood ladies living beyond their means. My impression is they may like the security of more money.

The old-fashioned jewellery is probably not being worn by them, even for Lady Middleton's dinner parties at Barton Park.
If the Georgian jewellery is old-fashioned, it maybe in the heavier styles of the earlier C18th with pearls, sapphires and garnets were the most popular stone of the C18th.

By around 1800, lighter styles of jewellery were more popular in fashionable circles such as a single rope of pearls or corals. Although such stones as garnets, emeralds, and other coloured stones did remain popular.

How galling for Elinor to meet John when she is selling her mother's jewellery, something she wouldn't be obliged to do if John had set an annnual income on her mother !
Meanwhile, he buys a seal for Fanny.
Then, she must listen as John lists his expenses- purchase of the East Kingham farm, Fanny's greenhouse, large purchases of linen and china to make his poverty clear.

Below is a link to an article on Regency jewellery on how jewellery fashions were changing.