Lower league management advice needed for Trials/Contract offers (NLS)

So I’m offering a bunch of trials to free agents to see who sticks and who doesn’t. I nail the shortlist down to about 5 players in each position, as soon as I offer a contract, another 5, 6 or 7 teams all do the same. This can happen with up to, like I said, 5 players in a single position. So I’m really struggling to get bodies in. Is there a way around this or something I’m doing wrong?

Cheers all!