I emailed help@tivoliaudio.com about the problem. Their suggested remedy:

It is pretty common over time for dust to collect in and around analogue tuners. Dust can cause drifting, scratchy noises while tuning, and difficulty finding stations. Regularly exercising your tuner will help to keep it dust free.

To exercise your tuner you will want to turn the radio on and turn the volume all the way down. Turn the tuning knob to the far left and then to the far right. Continue to vigorously turn this knob back and forth for two to three minutes. This will loosen any dirt or dust that may be lodged in your tuner and will recoat it in grease.

Success! After several minutes of vigorous turning, our radio’s tuning problems went away. Fingers crossed it stays that way.