This conversion seems only be possible using DesignCAD, apparently the only program capable of working with both .dc and .dcx file types. If this program cannot convert dc to dcx, then it is likely that no other program can.
Convert DesignCAD CAD file to DesignCAD file.
We found one software available in the database for this conversion.
Microsoft Windows software - convert dc to dcx on Windows
A 2D graphic CAD program from IMSI
Convert from dc file format
The table with program actions contains information about what each program is capable of doing with their files. This may be a good pointer for further searches for certain file conversions and help you find the converter you are looking for if our search result is insufficient for you. However, it is far from perfect and may sometimes show results which are not really usable because of the nature of the software handles the files. So a conversion that appears as possible, just because of matched actions, may in fact not be possible.