Once you have learned to mimic one accent you can quickly master another.
Source: I did this myself in Brazil when learning Portuguese. The words that were hard to pronounce I would say like I was from Rio. It was easier to mimic the accent from Rio because it was different from what I was hearing every day. If I were actually in Rio I would have probably found it easier to mimic a Paulistano accent.
I actually have been thinking about something similar for a while too. In my case, I thought that British accent would be easier for me to mimic. I haven't actually tried that out yet and now seems to be a good time to do so. Would British accent be a good goal? Out of curiosity, has there ever been a case of someone having a "too thick" British accent that would make it hard for American to understand?
If you have friendly native speakers, you may be able to get them to exaggerate the sound. E.G. the difference between thin and then is that the 'th' in then gets voiced (your throat vibrates) and the vowel is different. 'World' has an extra syllable (wur - ul - d) vs. 'word' (wur - d). I'm less sure about your issues with the others, save that certain sounds probably don't exist in your native language.
And those you just have to study & practice. I know how hard it is, though, having learned such sounds as 'ng', the 'eu' vowel in 'feu' (French - fire), glottal stops (yes, those exists in the English word bookcase, but...), the Japanese 'R'-group sounds (they're not 'R's or 'L's as we know them).
A slight Southern American accent sometimes does cause many Americans to perceive someone as more trustworthy, and some Americans do find a slight Southern accent to be very attractive. Though, be warned that a Southern accent may cause some Americans to perceive you as more stubborn and slightly less intelligent.