You mentioned you were considering living in or near Stuttgart, but your question title asks about Baden-Württemberg.
First note, that there are at least three different dialects in Baden-Württemberg, that I know of from experience of growin up there. These are
- Baden and
Since historically Stuttgart has been the capital of Württemberg, which correlates loosely with the borders for the Swabian dialect, you naturally find the swabian dialect more distinct in residents, especially older ones.
From my experience, @miep is entirely correct pointing out the difference between pronounciation and dialect. The Swabian dialect has a quite unique vocabulary, and a lot of words stem from French. In this dialect, all standard grammatics from Hochdeutsch apply, but speakers tend to use them cautionless.
For example, the sentence
Ich habe mehr Geld als du.
is easily botched by any Swabian into
Ich habe mehr Geld
which is wrong by all means. Because Swabian speakers understand they are prone to committing this mistake, they overcompensate a lot and write e.g
Wie ich nach Hause ging, sah ich ein Pferd.
Als ich nach hause ging, sah ich ein Pferd.
This was just about the grammatical structures, let me tell you about special Swabian vocabulary. Just have a look at this wikipedia section. Not that there is no uniform way to write German dialect, so when you have to resort to written German, the only way to go is Hochdeutsch, and every German writer will use it (plus a little local botching as shown above).
The biggest problem for speakers might be the pronounciation, though. Aside from the mentioned regions by other commenters, you might be hard pressed to find a Region in Germany where standard Hochdeutsch is spoken.
The three pronounciation I mentioned all do some ellipsis of vowels at the end of words, and handle at least the pronounciation of "st" and "sp", "sch", "ch" and "t" wildly different.
I can speak here mostly about Swabian. Swabian speakers are aware of their pronounciation, and as such, when they try to speak Hochdeutsch, they overkompensate also in their pronounciation. This leads to even more wrong pronounciation.
I have read this slightly humorous article, and if you understand German, you might find it exhilarating.
Please not that most younger Swabians do not excessively speak or pronounce Swabian. I just recently have been on a trip to Alsace, and I can definitely confirm that Elsässisch is not understandable to a German speaker, while every German I have met was able to understand (although also to recognize instantly) Swabian pronounciation.
Good luck in your decisions.