After I click checkout, I am asked to enter my shipping address. When I click continue, I get the error message "Error:  There was an error while processing your request, Please click  to start over." every single time. How can I check out and buy my item?

log out then try closing browser clear cookie like mentioned above, reboot your PC and try again
 
You are buying from your part of the world ?
 
North America or EU or other ?
 
Which browser ?
 
Try another browser
 
https://www.evga.com/about/contactus/
 

Last time I just used PayPal after my CC failed on a very small order I might add, used same CC through PayPal -- went through smooth
 
Might have been the difference between using Rd instead of RD for Road ...... its the CC company / validation system

Same thing happening to me and I'm pretty sure by the time it fixes itself the item i want will be out of stock. 

I see this sometimes with my business clients accounts. Most of the time it is not their pc nor browser issues, it is their end because they have screwed up entering information on either their paypal account or their credit card account.  Names, billing addresses, shipping addresses MUST all match EXACTLY on all linked accounts.
 
You cannot have your credit card info say James Smith, 1234 Main Street but your paypal account saying  Jim Smith, 1234 Main St. without a processing error. double check your account to ensure that all your info is identical at all processing locations :o)

 
Hope you got it sorted by now.