Illustrator adding lines to my dwg file

Hey, I was running into the same issue and played around with my file a bit. The solution Im providing is an assumption of whats happening to you, and also an assumption in part of how AutoCAD works.
I find that most contour maps come in as splines. Splines are pretty troublesome when exporting to anything (so are ellipses). I think this is because of how those curves are generated, there may be backround data and exporting/importing the file around breaks the association. Generally speaking, and from my experience, no one uses splines or ellipses in AutoCAD for this reason. Check and see if your lines are polylines or splines in the "Properties" tab of Autocad. There will be a little dropdown at the top of that window when you highlight one of your contours. If you're on mac you may need to google where that info is.

If you have a polyline; select them and make sure the "Closed" option is off. You may be able to select multiple at the same time to turn it off if it's on.

If you have a spline, select everything and run the "Explode" command. This will make the Splines into Polylines. This fixes the import issue but comes with some downsides. Every spline is made up of thousands of vertexes, so the file load gets intense. It also makes it very hard to edit your contours, should you need to interpolate or propose new ones. This is only really and issue at larger scales.

There may be a better answer out there for you, but if this is a one time thing, you might find the most success just running with this solution.

Does Autocad enable selected lines into a polyline? If so, then export. Hope that works.

Is your AutoCAD layout a flat 2D design or 3D? Generally speaking, AutoCAD doesn't tend to generate artwork that is friendly to vector graphics applications like Illustrator. The artwork tends to import with many of the lines broken into individual path segments. Rather than create closed shapes AutoCAD likes to create only enough lines that can be drawn on paper by a plotter.

You'll probably have to experiment saving copies of your AutoCAD layout using different settings in DWG or DXF format.