Is it my imagination or does the water get all churned up and visibility decreases around the new moon? It may be a coincidence, but I have noticed that every time we have ever had some viz issues or swam against heavy currents, it is around the new moon time of the month. Obviously, the moon affects the water, but I haven't noticed it so much at the full moon.

The moon is the major forcing of the tides, and the tides are a great source of current and thus stirring up of bottom particles. Once every 28 days the moon gets closest to earth, because of its elliptical orbit around the earth; the tides during this time are slightly larger than at other times..the perigean tides. Also, during full moon and new moon, the sun and the moon and the earth are all lined up, with both the moon's and the sun's tidal forcing being additive; these are the spring tides, and are the largest during each 28 cycle. And once in a while, 3-4 times a year, the perigean and the spring tides coincide.

There is no known difference between the new or full moon's spring tides, so I'm not sure what might be causing your observation that the viz/currents are worse at new moon.