Am I correct in assuming that if I have generated a 2048 bit dhparam.pem file which I pass to my web server, I am not using one of those "same few primes"?
Yes. And even if you did happen to be sharing a 2048 bit prime number with every other server in the world, it would still be essentially impossible for the NSA to crack in our lifetime (without massive advancements in quantum computing at least). 1024 bit is feasible for a powerful nation-state; 2048 bit is not.
Quick question - how many more times infusible is it? The article mentions roughly 7.5 million times from 512 to 1024, what is the rough jump required for 2048?
My math fu is not up to this at the moment.
2048-bit keys/primes are around 1 billion times harder than 1024.