There are a few threads about this already, but basically yes you can use the GI Bill as a full time student while in residency. You will get an additional 1034(+150 if you use the kicker)/month for 36 months non-taxed income.
You need to go to the VA website
www.gibill.va.gov and fill out the Electronic Application Form and submit it. Then if you are still AD then you need to print it and take it to your ESO to sign and mail it back to your regional VA office per the website.
They will then mail you back an approval form and follow in information. You will have to get your residency program to fill out the enrollment certification form(VA form 22-1999) and submit that back to the regional office. This must be done once a year.
When you are filling out the EAF under the heading which program you are applying for: Enter the GME office address and for course of study list what residency you are accepted for. You need to send the EAF in about 6 months before your projected start date. Then do the enrollment certification about two months prior.
To all this is roughly 43200 non taxed income over 36 months for 1800 dollars. The best return you will ever get for your money.
For Navy folks:
If anyone did not sign up for the GI Bill you need to submit a change to the board of correction of naval records(BCNR). via form DD form 149
http://www.npc.navy.mil/CareerInfo/RecordsManagement/Corrections/BCNR.htm
On it state that you were improperly counseled regarding your eligibility for the GI Bill.
Non-Navy Im sure your respective service has some similiar authority.