Is your GoDaddy account still able to be logged into (your hosting account, not the forum)? If so, then you need to export your forum database and save it. The bad thing is... you will probably have to use phpMyAdmin... and it's notorious for timing out. Once you have successfully saved that DB, then you simply import it into the DB for the site you are setting up on the new hosting and make sure that the credentials match. If you have already created a new DB for your new hosting site, you will need to drop that DB and then import your old one into it (making sure the config.php credentials match for the new DB).
If you need assistance (and the GoDaddy account is still valid) I can get with you via TeamViewer and you can log into your GoDaddy account and we can see if we can export that DB for you. Just let me know and we can start a convo. I'm usually up during the late night (US Central time) as I am
#1 - a person who has worked most their life during vampire hours
#2 - chasing after DSO captures due to my niche
BTW, if it's as simple as I think it is... there would be no charge. It shouldn't take that long to do and I can be working on other stuff at the same time.
Honestly though... if Nexcess is fully managed... they should be able to have done a simple panel -> panel transfer into their hosting environment.