Presenter Law received her portrait via the charity's fundraising director Beth Marsh, while Mr Matthias joined the show on the phone.
She said he had "caught her best side".
"You've caught the very witchy-looking hair I get this time of the morning when you get up," she said.
The picture was modelled on a photo she had taken with Tony, an ox created for BBC Radio Oxford by Oxfordshire-based yarn-bomber Yarnsy.
The artwork was sold for £2,800 at auction for the hospice in September.
"I've never seen a portrait of myself quite like that before," Law said.
Mr Matthias said he received multiple orders for his art from "all over the world" since going viral.
"They send such lovely messages about the joy and the laughter that it brings," he said.
"We advertise it as 'terrible', I know my ability in art and it's not very good."
The project is in aid of Beth's Bursary Fund, which supports palliative care nurses with training and education.
It was set up by the family of Beth Foreman, who died aged 32 in 2016 and was cared for by Sobell House nurses.
The project has featured original pieces by celebrities including Dame Judy Dench, Mel Giedroyc and Mark Gatiss.
All postcards will be available through Sobell House's online auction between 1-25 May.
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