I am finishing my MA in Teaching and will be a trained English teacher (with teaching credential) and TESOL certification.
So you would go in at the lowest level of the teaching profession, alongside new graduates (assuming that you can get your qualifications past nostrification and accepted by the school). That means you will be taking home a little over 2,000zl per month. To put that in perspective for you, a liveable three-room flat (i.e. one small room for whichever of your two kids are the same sex, a tiny room for the other kid, and a living/dining room which becomes a bedroom for you and your husband) in Warsaw will set you back at least 2,500zl per month. After five or six years in the profession you might be able to get that salary up to a level where it actually pays your rent.
I would prefer to work at a public school full-time and take on students for private lessons.
Private students can pay well but they are not something which one can rely on, only treat the money from them as 'extra cash'. You need a regular, solid gig to make sure that the bills and rent are paid in full and on time: a state school will not pay you enough to do that.
Am I dreaming to think that I can support my family doing this?
Yes. Simple as that.
My husband is perfectly capable of working, but has no 2nd language background and will be extremely limited in his choices of work.
There is very little work out here for people who have no marketable skill. From the sounds of it your husband would be limited to call centre work, which is soul-crushing and pays even less than being a state school teacher. Your combined net income would be maybe 3,500zl per month and that is low by Polish standards, especially for a family with three kids.
I have a family (3 kids and a husband).
What are you planning to do about their schooling? Polish schools are run in the Polish language and provide very little language help to kids who do not speak Polish. You would in essence be setting your kids' education back by a couple of years (i.e. instead of being ready to go to college at 18, they'd be ready at the age of 20). Do you really want to waste two years of their lives? Yes there are schools run in English, but the fees of those for even one kid would be far beyond your reach.
Personally I'd suggest that you stay in the US until your kids have flown the nest and then look at moving to Poland.