Will a TV antenna pre-amp reduce channel dropout?
An amp will probably make things worse, not better. Modern TV signals are digital, meaning they are compressed. But their new compressed nature also makes them highly susceptible to interference (such as your fog). 

In older analog TV signals, the problem was often a weak signal. An amp could help reduce that loss between the antenna and the TV. But digital signals are often made worse by amps, because they introduce a small amount of interference. As such, you lose information and, thus the signal entirely (digital cannot make use of partial information like analog could). 

I had to set one up for my mother and found that the amped antenna performed worse than an un-amped antenna. The only time you generally want an amp on your line is if you need to make a very long run of wire (if you're considering it, I'd just buy a separate un-amped antenna for the additional TVs).

Something else to consider might be an external TV tuner. TV manufacturers often skimp on the ones they install. I've even seen some that work with your wifi so you can get a solid signal in one location and then broadcast it over your house to streaming devices.