After looking through help files and forums, I finally came out with the solution to my problem (with the help of many online contributors of course). The solution, as expected, has to do with the rest ring that eagle sets.
Initially the rest ring was set at a min value of 10 mil for my DRC file. I lowered the value of that to 6 mil to avoid overlap on the header.
NOTE ON THIS: BE CAREFUL WHEN SETTING MIN HERE BECAUSE EAGLE SETS THE MIN FOR THE WHOLE BOARD, SO OTHER COMPONENTS WILL ALSO BE AFFECTED. FOR MY CASE, EVERYTHING CHECKED OUT GOOD, BUT TAKE CARE IF YOU USE THIS SOLUTION.
I think it would be more convenient if you could individually set the rest ring size for specific components since you have components of different sizes on the board, however, I don't know if eagle has such capability. Please correct me if I am wrong.
Anyways, hope this helps someone else out there. I spent quite a bit of time trying to find a better solution than just modifying the rest ring size for the board, however, I wasn't able to find anything else.
Also, for those who didn't read the comments on this question, here is another resource for anyone who has stumbled here with a problem with Via diameters: VIA DIAMETER PROBLEM
To add to the solutions. I re-posted this question one element14 community forums for Eagle. Here is the response I got.
Best Answer
Enable 'show grid' and check whether there are grid points on your pads. If not, the autorouter can't reach them. Solution: select a finer (routing!) grid.