Can someone with a VISA Type "F" convert it to a VISA Type "B", without leaving the country?

You cannot change non immigrant visa categories while in the country. You be able to change the reason for your extension of stay to one based upon working for company.

Where was your non-f visa and extension of it issued.

Visa F -  Organization A “work permit” - canceled after contract end with organization A then changes to
Visa F - Organization B “stay permit” (where husband is working)

The permission to stay is until next year.

"Permission to stay in Thailand under Section 15 until not later than xxx 2022"

No need to "change" the visa. The only requirement - in regards to the type of visa - to apply for an extension for working is to have a non-immigrant visa, the type of non-immigrant visa is not relevant.