I currently have Adobe Acrobat X, can I convert DWG & DXF file to PDF by installing additional plugins to support these file types?

Illustrator can usually open DWG & DXF files and can save to PDF, although you may get better results exporting a PDF directly from your CAD program.

First you have a very old version of Acrobat which is no longer supported by Adobe since the end of November 2015.

Acrobat converts to PDFs from most other file types by calling the application that creates or reads that file type and uses that program to print that file through the Adobe PDF printer. A few very common file types like txt and jpeg are builtin to Acrobat. Acrobat comes with plug-ins for MS Office. MS Outlook, AutoCad, HTML

I would check with the vendor of the program you used to create the file or can read the file and see if they have a plugin.