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.
EAGLE has no say over the outcome of the thickness of the actual copper layers themselves, since you'll be etching the board yourself and you're the one picking the copper board stock. However, you can increase the size of the annular ring to give yourself more room to drill a hole close to the center of a PTH.
Using the Info tool (the button that is the letter i), click on the PTH you are interested in and adjust the Diameter from auto to another value that is greater than your Drill size. Click Apply and visually inspect the hole - if it's still not as much copper as you want, increase the Diameter again until it's as thick as you want.
If you want to do this to all your PTHs, just use the Change tool, go to Diameter, and click the ... option at thebottom to set a custom value. This would ideally be the Diameter you chose when playing around with a single PTH. Then just start clicking on the other PTHs you want to update them.
This will only work, however, on PTHs you defined manually on the board layout. If some PTHs are part of the part's package, for example, you will have to edit their Diameter manually in the package itself, not in the board layout.
Best Answer
This is a one-off prototype, so doesn't need to withstand end-user mechanical abuse. I would probably trim the leads a bit, then set the ends of the leads on the pads, using the holes to align them. Now use solder blobs to hold the component in place.
The leads aren't going through the holes, but the ends are sitting on top of them. The solder guarantees a connection and holds the part in place. The part will be held much more weakly than if the leads were going through the holes, but for testing your circuit it should be good enough. If you really need more mechanical strength in your prototype, glob a lot of hot glue around the pins extending all the way up to the bottom of the part.
Before you forget, go into Eagle now and fix the hole and pad sizes for that part in your library. This can be easy to forget before the next revision when you're knee deep in changing other parts of the circuit.