Making paths in Illustrator - how to fix the cut segment in middle?

For my part you are welcome, ipeka.

Have you tried our suggestions?

ipeka,

Have you tried to complete/unite the incomplete/split counties (or what they are called (over there)) by using the Shape Builder Tool (ClickDragging over both brown and non brown parts belonging together to select)?

https://helpx.adobe.com/illustrator/using/creating-shapes-shape-builder-tool.html
https://helpx.adobe.com/illustrator/atv/cs5-tutorials/creating-complex-art-with-the-shape-builder-tool.html

It seems that some paths are not closed.

From this point onwards, I don't have a UI for English environment, so I don't know the exact menu item name, but I hope you understand the nuance.

Display the Document Information Panel from the Window menu,
From the panel menu, uncheck "Selected only" and turn on "Object". Then you can check the number of open paths on the panel.

However, this alone does not tell which path is an open path.
So I wrote a simple script, so please try it.

var sel = app.selection;
var count = 0;
for(var i = 0; i < sel.length; i++){
    if(sel[i].hasOwnProperty ("pathPoints")){
        if(sel[i].closed == false){
            sel[i].closed = true;
            count ++;
            continue;
            }
        }
    sel[i].selected = false;
    }

alert(count + "  opendPaths corrected.");

If you select all the paths that are likely to have problems and execute the script, the open paths contained in them will be converted to closed paths. Only the processed paths are selected and terminated.

Before running the script, make sure all paths are highlighted in layer color. If you have a mask object, it can be difficult, so you might want to do it twice with the "white arrow tool", half each.

I think Google knows how to execute the script. Good luck.