I just completed redoing the complete set of headliners on my '81 and '83 DeLoreans and found it to be much easier than I had expected. I would suggest replacing the entire set so that you will have a complete match that looks great when completed, and should last another 8 to 10 years. By doing a complete set, you can begin on the two center sections first since they are relatively flat and then do the doors last after you learn the technique for installing the liner.
The original headliner fabric is backed with an "open-cell" foam backing, that simply disintegrates in the heat and turns to dust, leaving the fabric dangling. The original fabric was glued to the pressed fiberboard backing with a contact cement that is sprayed to both the foam and the backing, allowed to dry completely and then pressed together.
When removing the old headliner, use latex gloves and an old sponge to rub the old disintegrated foam from the backing after pulling off the fabric. Be careful to keep the smooth surface of the backing. I had to repair a section of one of mine using a light layer of fiberglas matting and expoxy coating. to fill the gaps. Use caution in removing the backing from the door and top of the car. Use a head liner removal tool or two putty knives on each side of the plastic retainers that hold the headliner in position. You also need to remove the visors, and light housings.
Go to a local automobile upholstery supply house and you will find many shades of grey headliners for your selection. One large supply house that I used is "Keyston Bros" that has warehouses across the US. In the mid-west they have a warehouse in Dallas, TX (214)742-1875. They had the regular "open-cell" foam backed headliner for around $7 per yard (54" wide), and a higher quality woven fabric backing that was much more durable and not likely to disintegrate that was $12 per yard. This material did not have the color selection but I found one color #HL 2117 that was a light grey with a touch of metallic thread that really looks sharp. Use the best 3 M Headliner Adhesive, that runs $12 per can.
You only need 1.5 yard of the 54" wide material, and you get 2 "T" pieces for the middle sections and 2 pieces for the doors that are cut between the T sections. After removing the old headliners, lay them out flat on the back of the new headliners and chalk out the outline and leave about 1" around the outside of all edges. You will see that extra is needed for wrapping up over the edges of the door frames when they are installed in the vehicle.
Several folks suggested that I take the fabric and backing to an auto upholstery shop for spraying of the contact cement by a spray gun. It is most important to get a complete coverage on both the back of the fabric and the backing material. Let it dry completely and then begin by placing the fabric in the center of each section and working it carefully to each of the edges, stretching and pulling all wrinkles out as you move to the outside edge. leave excess to overhang that will either fold over, or be glued to the door edges for the two center sections. If you don't like the way it comes out, just lift the fabric up and stretch it into position. Once you like the look, then press it hard into position. This step is best done by two people, one holds the fabric above the backing, and the other works the fabric from the center to the outside edges.
Trim, and notch the fabric as it is folded over the back of the door sections, and trim the center sections after they are cemented to the edges of the door opening.
Use a can of adhesive solvent to clean off any excess cement off the fabric and any overspray, such as your hands. Clean up after it has dried.