When gas is flowing through the meter, I hear a steady, slow chirping sound (perhaps a chirp every 5 to 10 seconds). Is this an indication that the meter (35 years old) is coming to the end of its life? Should I be concerned about the chirping?
The noise is likely caused by a moving part rubbing its bearing or keeper. Whether or not it's a problem will depend on the meter.
You should contact the utility, and let them decide how to proceed. In most places the utility owns the meter, and is responsible for maintaining and/or replacing them.
Yes, call the utility & they'll have it swapped out in maybe 10-minutes & they'll probably be out next day. Your concern is that they can't accurately charge you & will estimate your bill with a wholly illegal & unwritten surcharge.