Yes, export an .fdf file from within Acrobat Pro.
Create a new .pdf with the fields you want. Spell them exactly the same.
Import the .fdf to populate the fields.
Also, inside of Acrobat, you can export the fields from one or many filled out forms to report.csv and open that in Excel. Delete or filter as you wish. They don't have to be open when you export them.
I only have Adobe Acrobat Reader DC - if there is no other way then I'll have to upgrade
You will need Acrobat DC Pro (the only version that Adobe is currently selling or selling a subscription for). Acrobat Reader is literally only for reading, and filling out forms.