Expect significant table variation. Usually of a very inconsistent nature.
For example, Ryujin's answer hinges on 'The dominating creature chooses your action.' effectively being written as 'The dominating creature may only choose one action for you to take on your turn'. Which it might be intended to be.
And if it is limited, does anything stop you from taking as many free actions as you want since Dazed lets you take free actions? Well, dominate says nothing of stopping you from taking any actions you want to, so apparently so unless you allow the dominated creature to choose actions.
And since free actions include some critical things for fun at the table, like talking, it's... a bit of a quagmire.
Personally, I think the intent is for the dominated creature to pick your actions, with the normal restrictions for dazed, and note that you can't take infinite free actions _anyways_. So, sure, it can have you drop your weapon and walk away from it. It could also have you hold your longsword in two hands (free action) and charge someone.
Or, if a dominated person got stuck in a literal quagmire that had a rule 'Creatures caught in this quagmire may make an Athletics check as a free action at the start of their turn or they're immobilized', a dominated person should in theory be able to make that free action, then take a move action out of it, rather than being perpetually stuck because them's the rules