I asked my landlord to register me online at the Immigration website, and then mail me the documents that proved my registration. She sent that, and one or two other things related, unsure of specifically what they contained seeing as all the text was in Thai.
I went with those, my passport and my E-Visa approval copy to Phuket Town Immigration. They have a little shop there that can scan, copy and take photos of you for a small penny. Took face photos and made copies of my passports ID and current visa pages.
Then I went into Immigration and submitted everything together with 600 THB, had to wait around 30-40 mins and they sent me off with two copies of stamped residence certificates. The bank asked zero questions about anything when they saw the certificate.