Free will is about intent, IMO. The act of "picking up" a coin and "picking up" the entity are two different things.
But, my understanding is that, simply, touching the coin gives the Fallen the opportunity to start tempting you to use it. In fact, I think that's kind of the point. Even Michael, a Knight of the Cross, refrains from touching one. But then again, Michael knew what he was picking up. But, I'm sure, the coins are purposely left lying around so that unsuspecting people pick them up. They also don't know what the coin is, but the act of picking it up, allows the angel to begin tempting them.
A denarian coin can't remove your free will. But it can keep working away at you until you have none left. Basically, it convinces you to choose it over something else until it imprisons you in a cage of your own free will. Which reminds me of a very a good quote that I read somewhere that I need to find...
What were you hoping for? To just drop it as soon as you realized what it was without it affecting you? I'm not sure that that will work...but that's just IMO.