Can seat availability for a flight be checked without going through the booking process?

I was looking at summer flights (mid August) JFK-AUA and am surprised that JetBlue is much more expensive, and the planes are at least half empty. I was curious if the flights were likely to decrease in price and wanted to easily keep an eye on availability.

There are two distinct questions/issues you're dealing with.

The first, assigned seats, is something that you can only check by going through the booking process.

It is worth noting, however, that the number of seats assigned is not necessarily an indication of how many seats have been sold or what the future pricing will be. Strange as it may seem, there are a lot of folks who book tickets and do not assign seats for whatever reason. If you are wanting to look at availability then actually looking into the number of seats for sale, not just those assigned. There are a number of tools, paid and free, that provide some insight into such data. I've written one of them that can display the number of seats in the various seat buckets similar to this:
Export from Wandering Aramean Travel Tools
And, finally, just because a flight isn't sold out yet doesn't mean that the airlines will be offering cheap seats in August. The advance booking window that airlines somewhat depend on for revenue prediction has shrunk dramatically. The cheapest seats aren't always (or even often, in some cases) available way in advance. If you travel JetBlue regularly you can buy now and then monitor the price and get a JetBlue credit if the fares go down.