Can I stay in the United States legally for a long time by going directly to a refugee lawyer after arriving in the United States with a tourist visa?

It is feasible in theory (feasible in legal procedure), but it is difficult to operate in practice. Not everyone can apply for a green card, nor can they enter the green card application process if they want to apply for a green card. There are rules and regulations in the lawyer's building. You can't mess around. Unless you are threatened by irresistible factors, such as imprisonment, death, persecution and so on. You may ask for it, and the Immigration Bureau will think that you stayed and were repatriated on purpose. This situation has been reported from time to time in the past ten years. Not one or two were repatriated.

Don't take the risk.