Thank you everyone for your help. It's not easy to figure out the procedures regarding visas. Here's an update:
The Thai Embassy in Hanoi sent me an email yesterday saying I had to talk with the Immigration Office here in Bangkok, but ended the email by saying it was impossible and I had to get a new visa (and I was more than welcome to come back...)
So I printed all my papers again + picture of receipt so I could try again.
I first went to the Department of Consular Affairs again this morning and luckily they had my paper from Hanoi that confirmed my visa! They just didn't notify me about it or answered my calls/replied my emails.
So the conclusion is:
You can get your Non Immigrant B visa transferred if you have the letter from the Embassy.
But without the help of Department of Consular Affairs is was impossible for me to get that letter from the Thai Embassy in Hanoi.