I didn't see it mentioned before, but has anyone else noticed that the server seems to go AWOL for ~5 minutes at a time every now and then?
Yeah, we know what causes it, and was aware the day before migration.
We have a couple of options available, but are still trying to get a baseline, so we know if any tweaks improve the situation or not.
Is it every time I view a page that's causing it?

Yeah, its all you rather fault, gayboy 
In layman terms:
Its the hourly backups
For those that understand, to get a point-in-time backup of the database (and it has to be point-in-time to maintain consistency), it is necessary to lock the database tables before the backup starts, and release the locks when its complete. In testing, this was taking less than 30s, which we thought we would get away with. Now traffic has ramped up, and an unexpected high number of users still browsing the old YaBB site, the server is working much, much harder, and its taking a few minutes. People then get fed up with waiting, so constantly hit refresh, giving the server yet more requests to deal with.
Bringing db to single user or lock mode is a must in all databases unless a new technology is introduced.. However, although expensive there are some alternative solutions like replicated/image backup but not sure this is a must or priority now..
just my opinion, stopping the old site and giving all efforts to the new one will be easier imo..And will decrease the server activity on disks ..
And from my experience, I always prefer to back up(sql) at late midnight hours , and during the day time, write the inserts, updates and deletes on sequential log files in case of a recovery necessity..