[Q] Can my family and I enter the U.S. on tourist visas and then apply and get I-526 approved and then adjust in the U.S.?
We do not recommend this approach. First, no one can guarantee that your I-526 immigrant petition will be approved within the period of stay granted under your tourist visas. Second, it's legally risky to enter the U.S. with nonimmigrant visa but with a pre-conceived intent to immigrate.
If you are based in outside the U.S., it is recommended that you get your I-526 approved and THEN take appropriate consular processing steps to obtain immigrant visa through the American Embassy located in your home country.