If you are sure that you qualify for a US Green Card and your appointment at the embassy is very close, then I would suggest that you apply for a new passport ONLY and ONLY if you think you won't have any issues doing that at your home country's embassy/consulate. The reason I am suggesting that is because the process of getting your passport back while your claim is still ongoing might be a complicated and tedious process. Once you get your US immigrant visa and before leaving Canada, make sure you let CBSA and IRCC know that you are withdrawing your claim and heading to the US.
I believe that being a Refugee Claimant in Canada should not affect your immigrant visa application to the US as long as you don't lie on your application. The only way having claimed refugee in Canada could affect your visa is if it was a visitor visa, as then they'll be worried that you might claim refugee or overstay your visa.