What's the exact text of the card?
D&D Home Page - What Monster Are You? - D&D Compendium
Well, if it says "use during your hero phase" it normally doesn't eat up any action and instead provides you with an additional move action for this turn, so you could do the following:
Attack + Move + Move
Move + Move + Attack
Move + Attack + Move
Move + Move + Move
Now it gets a little bit tricky. First i thought "well, just skip one of the move actions in the list above and you're fine", but i looked up the Dazed condition again and it simply states "You can only attack or move", not mentioning any actions being skipped. So yes, going by the rules, if you're dazed you should only be able to use the Potion of Speed if you don't attack (but you can run up to 3 times your speed instead! Woooosh! :D). If you attack, you're not allowed to move at all, so no potion will help you out. That's how i see it. ;)
You're right. On the marker it just says "Move or Attack" but in the rules it specifies that it's "Make a Move action or an Attack action". So first i agree, that you cannot move double when dazed. The potion part now is how one interprets the Dazed condition further. Does it forbid to move at all when you attack or is it just restricting your "base actions" but you can then through powers/items do whatever you please after that? Although the Dazed token surely seems to be just a short version of the rules and talks about the move/attack actions, i for myself find that the vague wording can be easily interpreted as "if you attack, you cannot move AT ALL, but when you choose to move then you can move even further through items/powers". After all being dazed means you cannot focus on more than one thing because it takes too much concentration to do so.
But after all, i think it's just another vague rule where you can do whatever you please as long as the people you play with agree... ;)