@Warren Bartram
Although there is an online service at https://convertio.co/doc-docx/ that does such conversions, there shouldn't be any need to go outside of what Word offers. When you say you've tried a "straight conversion", does that mean you opened the document, went to File > Info, and clicked the Convert button? When you did that, what sorts of formatting problems did you see?
Bear in mind that the conversion involves reading the binary data in the doc file and writing out the equivalent in XML notation in a docx file. That could result in occasional rounding errors in numeric data, which could then appear as slight differences in column widths, image sizes, and the like. I would expect the same result regardless of what method is used to do the conversion.
There are two more likely sources of differences. The first is that all versions of Word use data from the currently selected printer driver to position characters and other objects, so switching to a different printer can affect the layout -- even to the point of changing where some page breaks occur. Another source is that Microsoft replaced the layout engine during the update from Word 2010 to Word 2013. Documents that were created by older versions could show some differences when opened in later versions.
What this boils down to is that you should use the built-in Conversion routine and then carefully compare the printout of the docx version to that of the doc version. Where there are differences, decide whether they're noticeable enough to require manual corrections.
One other point, which may or may not be important to you: Word has always made a distinction between a template (.dot for Word 2003 and earlier, .dotx or .dotm for Word 2007 and later) and a document (.doc versus .docx or .docm). Using a document as the basis for new documents is an inherently risky proposition, because mistakenly using the Save command instead of the Save As command in the new file will overwrite the old "template" file.
