I’m currently staying and working in Dubai.
Is it possible to convert my Abu Dhabi family visa (sponsored by spouse) to Dubai work visa without exiting Dubai?
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Since you ask whether it is possible at all, this is more of a travel topic (how to do this would be relevant for Expatriates).
It is not possible to "convert" visas at all. You can obtain another visa; your previous visa might remain valid or be canceled. So your option is to get a new visa.
Generally it is not possible to obtain a visa for a country inside this country, only abroad. You might or might not be able to change your status inside the country from "family" to "work"; whether it is possible at all and what is the procedure you should ask in Expatriates.