You'll need to file these taxes regardless whether you are a PR or not since you are a resident in Canada. You have to declare your worldwide income in both your Canadian and US tax returns.
Double taxation shouldn't be happening. This sounds like an issue with how you are filing your taxes (not with the fact you are a PR). For any taxes already paid in the US, you need to declare these in your Canadian tax return to avoid double taxation. You will likely still owe some tax from a Canadian perspective since Canada has higher taxation rates, however if you file your taxes correctly, there will be no double taxation.
Maybe hire a tax professional to help you?