That's a bummer you have to connect to both. As I said, it's fairly uncommon here. Our first one was like yours but it died an early, painful death so we replaced it with one that only connected to the cold water.
I can't help you with the filter bit as I confess, we never checked ours. We just did the hot washes with vinegar or something.
I'd try at least one hot wash and see if that improves things.
(ps - I knew you wouldnt' have hard water! That's the perk I get being married to a geologist!

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(modified for a stupid mistake)