The antenna doesn't care how much power you put into it, as long as it's not so much that it arcs or melts.
When working with decibels they add. So if your amplifier is 5 W, that's 37 dBm.
If you put that into an isotropic antenna (one that radiates equally well in all directions) you would get a certain range. Your antenna is rated to have 16 dB gain over an isotropic antenna because it focuses the power all in one direction.
Adding 16 dB to 37 dBm gives you 53 dBm which is 200 W, so with your antenna you should get the same range from 5 W that on isotropic antenna would get with 200 W. This is ignoring cable, propagation, and other losses, but for this question we're just looking at relative power so we can ignore those for now. If you were doing an actual coverage study you would need to take all gains and losses into account.
Because you're now concentrating the signal in one direction you need to make sure the antenna is pointed directly at the antenna you are trying to communicate with, otherwise the signal will basically miss its target. The higher the gain the higher the directivity, and the more accurately you'll need to aim the antenna. This is likely the problem you're seeing.