"Is there a way to do this directly from InDesign?"
Totally. If you are printing this yourself***, then printing Postscript > Distiller is the way to go if you want to save your Booklet as a PDF, but you need to use a PPD that will properly define a page size. i.e. you must not select Device Independent as that writes the PS only for the objects in your file but does not include a page size, hence why it does not create a properly-oriented and aligned PDF.
The best way is to install the ADPDF9.PPD (available here https://helpx.adobe.com/ca/indesign/kb/add-acrobat-ppd.html) into the PPDs folder inside your ID install. This PDF will alllow you to select the proper page size or create a custom one for your own purposes. If your own printer happens to be a Postscript printer, you could use its PPD (like, I could use my Xerox PPD seen below just as easily), but there are limitations to that so I don't recommend it.
 
Once you have that you should be able to select the right page size to contain your printer spreads.
I recommend creating a Print Preset in InDesign for the size of paper you use regularly so you don't have to do it every time. For example, I print simple Letter-size booklets on 12"x18" paper (as I need bleed), so I have a Print Preset that ha sthat size, the settings for Crop Marks and Bleed, High Res Flattening, etc, etc, which looks like this: (adapt to your own needs, of course)
 Then when you Print Booklet, you can select this Preset. Check the Preview to see if you are getting what you expect, then Save the PostScript file and then Distill it to a PDF.
 
*** If you are NOT printing this yourself, you really should be Exporting individual page PDFs directly from ID and having your vendor impose the booklets properly. The issue with the Postscript workflow is that it is limited to Level 3 code and will necessarily flatten any live transparency and color management that would otherwise exist in properly exported PDFs from Indesign. That being said, if you ARE printing this yourself, this "flattening" is happening anyway, so you should get what you expect in print.
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