Whining Noise on Kenmore Ultrasoft 275

[smithta3]

I've got a 13+ year old Kenmore Ultrasoft 275 (it was here when we bought the house 13 years ago) that's started making a high-ish whining noise intermittently when the water is running anywhere in the house. Doesn't matter hot or cold, the noise persists. See linked video. I'd love not to have to replace the whole thing and I'd also love to avoid a $75 trip charge for someone to look at it. Any suggestions?