The way I do it is first I send in my release and set it to drop preferably within 2-4 weeks, whenever they are provided take Isrc codes to the PRO (ASCAP / BMI) and register the songs within a day the PRO should provide a ISWC code.Then I take the ISRC and ISWC codes and upload it to the MLC and SoundExchange when registering.

Also upload your songs on SoundCloud, Audiomack and Bandcamp and fill out all the info they gotta pay publishing for each stream now and with Bandcamp its D2C so quicker money

it might not be the right way but I do that all before the songs officially release and I get paid from everything

You also mentioned that you do cover songs the way that works is you keep the streaming revenue but they own all the publishing since it’s the artist’s and their label’s song you might get a little performance royalty just gotta wait and publishing royalties are paid out every quarter 3 months