Check Facebook Marketplace, Craigslist, or "Buy Nothing" groups for people who just unpacked from a move and are giving their boxes away. Often there are a lot of strong boxes (from moving companies) that have obviously been used once but have lots of life still.

Otherwise, I've had great luck moving multiple states away with boxes purchased at either Home Depot or Lowe's - about $2/box depending on exact size (and cheaper if bought by the 10-pack). Yes, it seems ridiculous to pay for boxes but when time is short, it's a lot easier to pay the $20 than running all over town than free ones.

EDIT: I'd also recommend sorting through the totes you already have in storage to see if you can declutter and get some space back.