Hey there, orangesmasher221!
I'm afraid it won't help. It's been awhile since your computer was manufactured, adding a 1 TB HDD might cause problems as well. Nowadays, HDDs come with SATA interface (3/6 Gb/s) and they often have compatibility problems with older systems.
Your Dimension 8300 is using IDE drive, so I'd suggest to try adding another IDE drive in there.
As for the BSOD error, check this knowledge base article:
https://support.microsoft.com/kb/324103
Hope this helped! Good luck!
SuperSoph_WD