Would the file sent to Dropbox still count against my form limits?

I believe you are referring to the Upload Storage limit and the Total Submission Storage limit which are Fixed Limits as explained in this guide — Understanding-Your-Account-Usage-and-Limits

Yes, even if the submissions are also being saved to your Dropbox account, it will still count against your account limit because it is still being stored in your Jotform account. To free some space, you can always delete the submissions in your form — How-to-Delete-Form-Submission-Data. I understand that you are planning on saving your form submissions in your Dropbox account but we still advise that you create a backup of your submissions before deleting it

How-to-Partially-Download-Form-Submissions-as-Excel-CSV-PDF

How-to-download-all-or-partially-your-submissions-file-uploads

Deleted submissions are irrecoverable.

Another option that you can do is to automatically delete your form submissions from your Jotform account as you receive it using the Auto Delete Submissions App

http://auto-delete.jotform.io/

Forms added in the app will have its submissions instantly deleted as soon as it is received.

I hope this helps. If you have other questions regarding this, please feel free to contact us again anytime.