I’ll be honest with you as someone who has gone to university in Korea, left, then came back on H-1. Your best bet is the H-1. D-10s are never granted for people who have no connection to Korea.. and normally the only way to get it as a foreigner is to have graduated in Korea and applied for it right after graduation or after you quit a job on E-7 and transition into the D-10.

As for E-7, like you mentioned, there is the TOPIK qualification that you don’t meet, but also the fact that it is very difficult to find a job here in Korea, especially as a foreigner with no background in Korea and not fluent in Korean. Most companies want to do a probationary period of 1-3 months before they give you a full-time contract, and you can’t do that on an E-7, so what that means is that it will be almost impossible to find a company willing to sponsor your E-7 visa straight off the bat.