First thing that comes to mind, if trying to dodge a single attack takes one of you 3 APs per round, facing multiple enemies will be really dangerous.
Welcome to combat that starts to look realistic!
A simple method would indeed be giving 3 action points at the start of your turn, and you can use all 3 or leave some in reserve for reactions later.
I would leave an ordinary defense (block) for 0 action points that may just be an armor roll for example. A dodge for 1 AP would be perhaps a dex save equivalent in 5e or the like. If the dodge was failed then they can make an armor roll. This makes the 1 AP spend quite useful as it gives two chances to ddfend.
If the above worked like i imagine it might, it might give players the choice to go all out and attack 3 times, or try and be more defensive. Heavily armored and slow players may be less likely to dodge than nimble players. And perhaps even certain options could bs made available for certain weapons (cleaving attack with a greatsword for 2 AP for example, or perhaps a riposte reaction with a rapier fkr 1 ap) to grant limited extra choices.
This is what that system would look like:
Modos RPG.
Ordinary defense, a parry, costs one action. Armor absorbs damage, so if you don't want to parry/block, you count on your armor to protect you.
All-out attack is definitely an option, but it rules out any chances to parry. This is where Olrox's concern comes in: if you're outnumbered, the enemy (enemies) can attack you more than you can defend. So you'd better have a plan.
Nimble characters, who can't count on their (light or no) armor to protect them, must instead count on dodges, positioning, and timing. As well as a weapon with some range.
For OP:
d20 rules are already broken up into actions: primary action, movement, and a reaction. I don't see why each of these has to occur on a PC's turn. Also, if that reaction can also be a parry/block, you might have the system you're looking for.