One of the qualities I find most important in food is the taste it leaves in my mouth.
However, the only term I have for this is "aftertaste" which tends to suggest something unpleasant, and has a subtly different nuance than what I'm talking about.
When I eat an Oreo Cookie, it is the memory left on my taste buds that makes me want the next one. That is the kind of aftertaste I am talking about.
Is there a better word or phrase for this?
This is what wine connoisseurs call finish.
Described by Total Wine as:
The last impression of a wine is the finish: The taste that stays on the palate after the wine has been swallowed. The length of the finish is the final indicator of the wine's quality. That taste can be short and crisp, or it can linger for a minute or more, continuing to unfold the flavor secrets of the wine before finally fading away.
I don't know of a single word for it, but the phrase pleasant aftertaste seems to be pretty common in recent decades.
Looking at some of the example hits, it seems to be used most in reference to wines and beers.
Savour (or Savor AmE.) as a noun is a :
a smell or taste, especially a pleasant one
-Cambridge online
However, I do not really find a problem with the word aftertaste.
For example, on the website World Food and Wine, under the heading of Describing taste and flavor....
Aftertaste is the trace, hint, smack, relish, savor food leaves behind.
and does not seem to have a negative connotation. The words "smack", "savor", and "relish" would tend to indicate the complete opposite.
From the Culinary Dictionary, the term seems to be neutral:
after taste – Taste which returns to the mouth after ingestion of certain foods and beverages.
I can see why you'd be concerned about "aftertaste" carrying a bad connotation. I agree with @Barmar, however, in that adding "pleasant" or "pleasurable" to "aftertaste" should cancel any bad connotation.
As you're looking for a word that expands past the sense of taste, you might consider "mouthfeel," which is "the tactile sensation a food gives the mouth."
http://www.dictionary.com/browse/mouthfeel
It isn't the most elegant word, but it does refer to what you're describing. It also doesn't carry any connotations, positive or negative, and it isn't limited to the "taste" sensation. You should be able to throw modifiers at it to make it suit your needs pretty easily.
Edit
On @lawrence's recommendation, I've added to my original answer.
I hope these two explanations will further clarify the term "mouthfeel" and how to use it. Both source articles have expanded information about the term and its various components. You can find a link to the source article at the end of each quote.
The term "mouthfeel" (or "mouth feel") may be used to describe the sensation produced when a food / drink is in the mouth or after it has been swallowed (as is the case with the drying sensation caused by astringent tastes). https://www.thespruce.com/mouthfeel-765730
Texture also plays a crucial role in how much we enjoy food. This has to do with how it feels to it — from first bite, to the swirl-around inside your mouth, to how it lingers after you swallow. https://www.theblot.com/food-addicts-glossary-mouthfeel-7713785
(An explosion of creamy sweetness) with a lingering hint/taste of chocolate.
(An explosion of creamy sweetness) leaving traces of chocolate.
(An explosion of creamy sweetness) followed by a nostalgia for chocolate.
(An explosion of creamy sweetness) followed by the faintest daydream of chocolate.
(An explosion of creamy sweetness) chased by a suggestion/hint of chocolate.
