Exterior of addition's foundation leaks

The driveway has not shifted but has sunk..

Why concrete sinks in the first place....

Fill dirt is almost always placed along side of house and garage foundations after the foundation work is completed. This fills in voids created during the foundation construction process. Rarely does a builder take the time to compact this dirt.

1: You will not be able to re-apply some 'good stuff' to make it work.

2: The addition has some issues of its own in that it had leaked prior to - what company did the addition - perhaps pulling the building permit might tell you who did the addition - if there is no permit on record - that will tell you a material fact - whether you have any recourse is up to State Law (with a name like rocky - I assume you are in the USA).

3: You might not have to replace the driveway - you might be able to have it lifted, this is known as Slab Jacking, Mud Jacking, Poly Urethane Filling - your driveway has not shifted but has rather sunken from the load on the dirt above - there might be other issues at play here as well - it has sunk on your addition side not the other side and you mentioned problems with the addition - a very big reason why you will want a professional to look at it and also why you want to look for those building permits in your city or county office. Again there might be avenues where the responsible party ($$$$) is not you; but then again state laws vary and Caveat Emptor is usually the motto when it comes to Real Estate.

What is typically done is a fill material is inserted underneath the concrete and into the soil to lift the sunken areas. What you will need to do is search for a company in your area that will perform 'Mud Jacking or Foundation Repair'. Mud Jacking is not a cure all for all foundation or sunken slab issues but it can be a cheap alternative fix for many problems.

Now a link for estimated costs as that is probably going to be important to you, typical costs can range anywhere from $400 to $4000 depending on the severity and the issues found as well as the company performing the service.