How do I purge Windows of my private data when formatting the drive is not an option?

Note that if you were on an AD Domain, domain administrators would have had full access to your computer anyway. The usual caveats about physical access, unencrypted drives, etc, all apply, so this is not real security but will prevent subsequent users of your computer from getting easy access to your data.

If you were not part of a domain, then the best you can do is create a new administrator account, and then delete your old account and profile from the new one.

Make sure that the recycle bin has been emptied. If the Volume Snapshot Service is running, delete any volume shadow copies by running cmd with elevated privileges:

vssadmin delete shadows /for=c:

Finally run the following command for each drive:

cipher /w:c:\

Where c: in both cases is the drive letter designation. This will wipe all free space, making it unrecoverable. See this answer for more information.

First, use a software to securely delete files like Recuva or Eraser.

Apart from formatting, you will have to rewrite the empty space with 0's or 1's like @Begueradj has suggested.

There are softwares which will just do this for the empty space in the drives like CCleaner.

Just run a 3 or 7 pass wipe to rewrite and securely delete any traces of your personal data from the empty space after your delete them.

What CCleaner does is create a file and start writing 0's or 1's till your drives empty space is filled. Then it deletes the file.

Good Read: Learn How to Securely Delete Files in Windows

I suggest you a naive, tedious but still effective methode:

When you purge the data (also from recycle bin), Windows marks the related hard drive space as available for future reuse but the deleted data remains there until you use that space. Thus, you may think of storing lot of data on your hard drives so that you can reuse those available for reuse spaces. This is what you can call simply as overwriting data. As I said, this method can be manually tedious, but it is still effective.

P.S.

Note that you can overwrite those deleted data with random data (sequences of 0's and 1's) Darik's Boot and Nuke (DBAN) in case you can clear a full hard drive.