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.

I would recommend taking a month or two to "practice" living on less discretionary income. If your rent is $2100 now and you expect a mortgage to is etimated at $3400, then pretend you're paying the full amount now (put the difference in savings). See how tight it feels to not have wiggle room for hobbies, restaurants, whatever.

If that feels okay, then do it again the following month but subtract yet another $1000 from your monthly budget to cover the unanticipated costs of first-time homeownership: repairs, maintenance, higher utilities (due to more square footage) or simply buying things like ladders and lawnmowers that renters don't need. Eventually these costs might decrease but by then your childcare costs may be a reality.

If both months feel like sustainable budgets to you, then go ahead and buy a house as planned. Otherwise, adjust your purchase price or increase your earnings or postpone your purchase.