You have discovered the disadvantage of using unusual fonts. You may try embedding the fonts when you save the documents, see the instructions here.

The ultimate solution is for everyone who need the unusual font to have it installed.

You are correct that when Word does not have the font, it reverts to the default, but it still may say the name of the missing font in the fonts formatting window.

I have seen this at the university where I work. For many years, the officially specified font did not even exist for Windows, only Mac. The vast majority of computers on campus were then, and are even more so now, overwhelmingly Windows computers. It drove us crazy. We just used Times New Roman and ignored the requirements. Doesn't sound like you have that option.

If you are working on things for customers that require you to use that font, buy it and consider it part of the cost of doing business. Pass that cost on to the customer, too. It's a legitimate expense.

On edit, I found it very quickly on a free download site using Google.