I don't think it's a feature of southern accents, per se.  Probably 80% of Anglo-Americans (I've heard speak) pronounce in this way.

There are some accents in the USA which are closer to UK English than others, so they may exhibit similarities to what you've stated...  But overall, I don't personally find this to be the case.

Americans do say "thee," just not in general colloquial speech as a "rule."  It's more of a tool to aid in comprehension or detail highlighting in specific situations/circumstances.

In American English, "the" is almost always "thuh" (thus, without the /s/), unless it's being emphasized.  The vowel is often shortened, as it's a function word and normally reduced.  Unstressed Syllables often see changes in both vowel quality and length.

The /th/ in "the" is voiced.

There is almost never any yod-linking, and the /e/ is only pronounced when it's being emphasized.  Even then, most Americans will not link with a "yod.".

This includes before vowels...

Example Sentence:  https://vocaroo.com/i/s1vT4erfI9NG

The apple doesn't fall far from the tree.

EDIT:  The + letter that starts with a consonant sound ( B C D F G...), pronounced like "thza"

There is no /z/ sound in there.  This pronunciation is usually a symptom of language interference that manifests in the accents of non-native speakers (and some bi-linguals, if their environment included heavy use of this pronunciation).

"th" in "the" is a voiced inter-dental fricative.  /z/ is a voiced alveolar fricative.  You cannot make anything which sounds like a /z/ from that articulation.  It's literally impossible without adding on a[n incorrect] secondary articulation, which is basically an extra consonant ;-)

Before a vowel sound, the usual pronunciation is "thee", often with a 'y' sound to link the two. So the phrase 'the apples' might sound like "theeyapples".

Before a consonant, it is usually reduced to a schwa, which makes the unaccented 'the ' in 'Give me the books' sound like a soft, short "thuh". In fact, in rapid speech, the vowel is barely audible at all, and might sound more like 'Give me th'books'.

NB Note that this is a question of vowel or consonant SOUNDS, not spelling. So we would say "thee" in 'the hour' (because the initial consonant is silent), but "thuh" in 'the universe (because the first syllable is pronounced "you").

However, we sometimes use an accented "thee"  pronunciation before a consonant to emphasise the superlative or definitive nature of something. For example, someone might say 'You are THE ..best..friend..ever!' slowly for emphasis, making the extended 'the' into a "thee".

NB These pronunciations may be different in different English-speaking parts of the world.

Thank you for the explanation, Nate.

It was this difference that I had in mind when I mentioned possible regional variations. ( The last time I answered a question about the "thee" pronunciation, I received a torrent of abuse from an American girl, saying how wrong I was. I wanted to avoid this happening again!)

Can I ask whether the preference for the "thuh" variant, even before vowels, applies to American English in general? Or are there regional differences within the US? I had the impression - again, possibly wrong - that this was more a feature of Southern accents.

By the way, I agree with you about 'z'. No native English speaker would insert a 'z' sound into the word 'the'. This is a non-native error.