First, you can always import Word files into OnSong and view them "as-is". You do not need to convert them to text-based documents to view the files. However, you will not have access to OnSong's formatting, transposition, and other features. So we would recommend importing the Word files into OnSong directly into OnSong using Dropbox.
If you want to move to a text-based format, it's best to do so one song or set at a time. The reason is that different files are created differently. The only reason to change the font to Courier or New Courier is to you can see the issue with font alignment and fix that in Word. The trouble is that when you author a chord chart in a word processing file, it's common to use spaces to align the chords above words below. When you use a variable-width font like Arial or Times, you end up using a lot more spaces than characters below. So when you import or extract that text in OnSong, the chords become misaligned.
Our recommendation is when you have time and are viewing a Word document in OnSong, open the Song Editor by tapping the Attachments Menu icon in the Menubar, and locating your Word file. Hold down on the file until the context menu appears, and select Extract Text.
(In OnSong 2020, tap on the text cursor icon to open the Song Editor. Locate the Conversion Toolbar at the bottom of the screen. Here you can locate the Extract Text button in the lower left if the imported file format is supported. Tap on the Extract Text button to have OnSong extract the text from the file.)
This will convert the Word file into plain text. What you may notice is that chords are not aligned above the correct place in the lyrics. Tap on the hammer icon in the Song Editor Menubar to open the Text Tools Menu. Tap on the Fix Alignment Spaces button. This will adjust the chords closer to where they belong above lyrics. You can then manually adjust the spacing before saving.