Moved WP Site to New Server, Now Cannot Log in

Have recently moved my WordPress from host gator to a new hosting company.

The first time I navigate to my /wp-admin page to log in, it already tells me that I only have two log-in attempts left. After I type in correct username and password a single time, it locks me out for 20 minutes for too many failed log in attempts. If I navigate back to the page (without attempting to log in) it will eventually bump that up to 24 hours.

This is happening to about half of the folks who log in. All are using different computers in different parts of the country. Have tried clearing cookies, resetting computer, resetting modem, purging DNS cache on my machine.

Thanks all for any possible solutions!