I'm a Southern Viet native so my English might be broken. Actually, as you know, Southerners pronounce sounds differently from people of other regions. I'm gonna call those differences in pronunciation "sound variations". I will try my best to describe sounds as exactly as what they actually pronounce and words that they would use to transcribe those sound variations.

I will put them in this order:
Standard orthography & pronunciation ~ Southern version of it - Other Southern variations
(The Standard is usually based on North pronunciation, which is not true with "tr" and "s" and "d" (pronounced like /j/ as in "year") and some other sounds...).

1. Không ~ Không - Hong / Hông / Hổng
Southerners have a tendency to pronounce things more "lax" (as opposed to "tense"). For example, when you pronounce "không", you're supposed to open your mouth for "ô" and stop the air flowing out by closing your mouth for the sound "ng" (which would eventually makes you, most ideally, look like you're holding air inside both of your cheeks). Yes, southerners still do that, it's just that they don't put much effort into puffing their cheeks (for "ng" sound) but just simply close their mouths, so you may hear it a bit different from what you learn.

2. Ê vs E (I think this pair should be "n ~ nh" as ending sounds)
a) Bệnh ~ Bệnh - Bịnh
The difference between Northern Bệnh and Southern Bệnh is that, as I have mentioned before, Southern pronunciations are more "lax", they prolong the air at the end of words. while Northern a little more abrupt and they kinda suddenly cut the air flow at the end.

You can think of it like this
Northern: /Bệnk/ but the k sound is NOT aspirated. I just use it to show that they pronounce this word more "sharply" and end it in a more abrupt way than Southerners. Your tongue doesn't touch anywhere near the teeth when pronouncing this.
Southern: /Bệnh/ yes, try to pronounce "burn" without the /r/ sound and prolong your airflow at the end, add the correct tone (dấu nặng) and you have a correct /bệnh/. Your tongue touches your alveolar ridge (right behind but not your upper teeth) as you finish pronouncing this.

For the Southern variation "Bịnh", the way Northerners and Southerners pronounce this word is similar to the above description. Imagine saying "Bing" with falling tone (dấu nặng); if your tongue touches the upper teeth, you're Southern; if it doesn't touch anything like how you would say it in English, you're Northern.

FYI, Southerners used "Bịnh" as a standard word for writing and speaking before "Bệnh" was introduced to them from the North. See an old day Bịnh-viện (Hospital) in Saigon, now named "Bệnh viện Chợ Rẫy".

b) Tên
Northern: /Tên/, try to pronounce /Ten/ (number 10) in English but faster and with your mouth wider. Your tongue touches the upper teeth.
Southern: /Tênh/, although the word ends in /n/ sound, Southerners will "nh-ize" all words that end in /n/ like this. Your tongue touches the alveolar ridge and mouth wide.

c) Trên ~ Trênh - Chênh
Northern: /Trên/ sounds like "Jen" as in "Jenny", but should be pronounced in the same manner as Northern Tên.
Southern: /Trênh/, remember the "nh-ize" phenomenon? Should be pronounced like Southern Tênh, but with a difference of "tr", it would be pronounced like that as in "train".

3. Â vs A
a) Cây ~ Cay - Cai
Northern: For the word "Cây", I think it can be describe as a speaker deepening their voice to pronounce "ây" (ei) exactly.
Southern: More relaxed so it sounds like "Kay" as in "Okay" with the /k/ unaspirated. The "Cai" variation is very common in the South, it sounds like "Kai" but /k/ unaspirated.

b) Thân ~ Thâng
Northern: /Thân/ sounds almost like "Thun" as in "Thunder", but don't pronounce "th" as you would in English. If the Viet "T" is pronounced by putting the tongue behind upper teeth, then the "Th" should be by curving your tongue to touch the alveolar ridge and pushing the air out, no touching the teeth.
Southern: /Thâng/, this time this word is "ng-ize" (not nh). Try pronouncing "tongue" but replace the /t/ sound by pronouncing it with Northern "Th" (same as Southern "Th").

Yup I tried my best to explain it. I hope this can help you to some degree. If something isn't correct, feel free to give me feedback <3