PMEG does not provide a theorical meaning for adi. What would be the most probable meaning of adi based on how other affixes are transformed into verbs?
The suffix -ad- points to actions, repeated actions, long activities, and habits. You will occasionally see ada and ade used to mean "continual(ly), habitual(ly)".
Adi fari is rare but means "keep on doing, continue doing" (in a fairly purposeful way):
...vi adis defraŭdi abundon da faruno... ...you appropriated over a long period a great quantity of flour... (From La skandalo pro Jozefo)
As for adi without a verb following it, I think I would interpret it as "continue actively" or "busy oneself". The word umadi already means "linger, potter around" but adi sounds a little more strenuous.
Li nin vizitadis. He visited us a great deal.
Li ade vizitis. He continually paid visits.
Li adis viziti. He kept on visiting.
La laboro ados senfine. The work will be unceasing.
Li adis en la riparejo. He fussed around / spent a lot of time in the workshop.