converting breezeway into indoor living area.. have concrete issues
If the existing concrete is an on grade slab without footings (and since it is sinking, that sounds likely), most localities will not allow that to be converted to living space. You need to discuss with your building department.
You may have to tear it out and install footings, stem walls and a new slab.
It will depend on the part of the country you reside and how deep the frost line goes.
Coastal South Carolina