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For the 12-hour format, could AM/PM be presented as am/pm, or at least at a smaller size than the digits?
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For single-digit days of the month, could the extra space be removed between the month and the day? (en-US region; please see the screenshot below).

Or are these things that I could adjust locally by editing a text file?
For #1, AFAIK there is no way to change this aside from a code change.
For #2, however, there is a custom date format you can set. If you have dconf-editor installed, you can edit the /org/ubuntubudgie/plugins/budgie-showtime/dateformat key to change this.
The value you would want should be:
%A %B %-d
** Note: you need to logout / login to see the change
I think this should produce the result you want for the date at least:

You can see the list of options here:
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Thank you! I’ll give it a try.
edit: Yes, that worked.
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