Many greenhouses give away pots. (People often return them, and the greenhouse may or may not actually have a use for them.) Sounds like 1-gallon pots (used for shrubs) may be the ticket, since your seedlings will do a lot of growing between now and then. I've been digging up countless clematis seedlings the last couple of springs, and putting them directly into 1-gallon pots, where they grow happily until the orphans are given away. Just stick a branchy twig into the pot for them to twine up.

How about pop containers from a fast food place. They are waxed and plenty deep, particularly if they are "supersized". Some are even plastic.

How about 2 litre soda bottles? You cut off the neck on top sort of just above where the label is glued on. Poke a few holes in the bottom for drainage.

Not sure about 2 litre milk cartons (the ones you mention that are essentially paper coated with wax). They might work really well but also might start to fall apart by spring. I guess it depends on how much they are moved around, exposed to sun, etc, because the wax sort of wears away and eventually they start to fall apart. They work great for starting stuff in spring, where they are only used a few months, however.

I know these aren't big enough, but in case someone is reading and needs ideas, I bought big red disposable cups and started my petunias in them. They are a bit bigger than beer cups, maybe 16oz. Of course, they aren't big enough for you to use because you want something more than 4" deep.

Regards,
Glen