Sorry I've missed the opportunity to answer the question about 전 and 제.
But I couldn't have come up with any better answer than your own. Well done!
Sometimes a lazy teacher produces an excellent disciple. ( ㅋ ㅋ , in English, l o l )
Now, about regional thing,
I'm a Seoul native, but my parents are from Daegu, so I'm bilingual.
Daegu people can't pronounce ㅆ.
They cook 밥 with 살, not 쌀.
I rest my case there.
Let me just tell you how you can soften ㅂㄷㅈㄱ ㅅ.
First, with ㅂ, think how you produce that sound using what part of your mouth.
Lips, right? No need to wisper because it's not the air that can change the sound but how firmly you're holding your lips has the key. Try soften your lips, or tongue touching your palate when making ㄷ sound, and so on.
계속 연습하세요, 화이팅! ( in Daeguan, 잘해라, 쫌!)