What can I do to get the Drivers Badge after meeting the requirements at my previous unit?

As long as your record in GCSS army reflects that you meet the requirement then you just have to get with your battalion master driver. Also if you have a DA Form 348 or DA Form 348-1 that you brought with you from your last unit that would help.

SSG(P) (Join to see) Since you have posted a couple of different questions in regards to awards I have looked at your profile. A suggestion - update your profile.
And from your photo and your profile, it looks like you have a Ranger tab and a Sapper tab, if that is the case, who did you PO?
In looking at your profile, I do have to question when you hit your three (3) year mark.
Next, did you PCS shortly after graduating from the Sapper Course?

First thing in regards to your GCM, check with your current S-1/HR group, after that contact your old unit. As far as the Driver Badge is concerned, make sure that you have the documentation required and send copies off to your old unit and ask if you were put in for one and if not request that they do. This would also be a good time to ask about the possible PCS award that you feel and appears to be well deserved.

Good luck in your endeavor and please update your information. You may want to also check your personnel record (whatever it is called today) and make sure that it is up to date - that is your responsibility and no one else's.

The Good Conduct Medal is easy, your current Commander is the approving authority using the information available in your personnel records (para 4-4, AR 600-8-2). See your S-1 to get the paperwork started.