Basically your NM permit is good until your Drivers License becomes a Colorado license. The address on your permit must now match the state that your permit is from in order for it to be recognized here. If I recall correctly, you have 6 months to get your license changed once you have taken up residence here.
What part of CO did you move to? I'm asking because as long as it's not Denver, you can open carry without a permit and carry in your car without a permit. If it's Denver, my suggestion would be to get your CO license right away and then start the process of obtaining a CO permit. In the meantime, just keep your NM permit and driver's license available in case you're stopped by LEO, and just explain that you're visiting from NM and whatever you do, don't indicate that you're establishing residency in CO until you've actually received your CO CCW.
Per CO Statute 18-12-213 (b) II, you have 90 days from the date of becoming a CO resident to get a CO permit. Residency is typically defined by the State that issues your DL, the license plates for your personally owned vehicles or where you're registered to vote. Take the class and apply now.