Best-quality copies of the show

Due to copyright issues which the studios and interested parties involved have never resolved, there has also never been any official releases of this cartoon to DVD or digital download. However, it is available for streaming on Crackle and bootleg copies elsewhere.

A service I have used in the past for hard-to-find shows has "Jeannie" available as a digital download (Mantronix). They say that the episodes are uncut, but unless Sony ever releases them from their vault, they remain cut for modern advertising-break lengths. I know this not only from the average time of each episode (ranges from 21-22 minutes) but also scenes I remember seeing on a couple of episodes are no longer there. Yes, I'm old enough to have seen this show when it originally aired (and I did watch it).

Here's the direct link to the Mantronix page for buying the series, though I do recommend watching it first on Crackle. It's "free" with ad-breaks, but you'll at least know what it's like first.
https://mantronixretrotv.onlineweb.shop/Jeannie_-_1973_-_The_Complete_HD_Studio_Animated_Series_(Digital_Download)/p6119874_20753506.aspx

A couple of us I Dream of Jeannie fans have added screen captures from the animated show on the IDoJ wiki-page if you want to have a look there as well:
https://i-dream-of-jeannie.fandom.com/wiki/Jeannie_(series)

One thing I have read about streaming versus other copies is that the streaming copy is often higher in resolution than other forms of media. So, if you know how to capture streaming video in whatever highest-quality form you can, that might be the best way to get the series, assuming you'd want to keep it. As a fan of all things genie, I did, but admit that I'm an outlier overall.