"Please note that relatively few people can begin receiving a benefit at exact age 62 because a person must be 62 throughout the first month of retirement. Thus most early retirees begin at age 62 and 1 month."
https://www.ssa.gov/oact/quickcalc/earlyretire.html
So you aren't eligible for benefits until June anyway.
SSA is very understaffed. Keep calling (either the local office or the national 800 number) and if you want, contact your Representative and/or Senators to sign a release and have their case work staff help you inquire about the status. I know the head caseworker in Jamie Raskin's office and this type of work is a big part of what they do.