Lawrence,
What you are getting is the standard hot-rolled product. Final rolling of the component, be it plate, flat bar, etc. is done at high temperature. What you describe (cleaner, almost bright) is probably cold-rolled product. The final processing is not done at high temperature, but you will probably pay a premium. Depending on the size of weld coupons you want (length and width) you could order cold rolled flat bar. Check out this link -
http://www.spaco.org/hrvscr.htm. You can Google "cold rolled steel plate, flat bar, etc." to find a lot of info.
Hope this helps.