I have personally been using Sensodyne Rapid Relief Mint
It cleans exceptionally well if that is good enough.
... and I like it a lot because when I was having carious tooth pain from a failed/leaking filling and could not isolate the pain to a specific tooth, I could actually feel this tooth paste get into where it hurt and sooth some of the pain. It also got my teeth very clean, and it tastes fantastic.
The first few times I used it, this toothpaste felt funny on my teeth almost felt like it was warm and burning a bit right in the spot that hurt; I did not like it at first. Then it started feeling better and soothing. It reduced my pain and sensitivity a lot.
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That said my dentist says it is best to get a non-whentening, Stannous fluoride, ADA approved toothpaste, with minimal to no Sodium Laureth Sulfate/SLS (foaming agent) because that irritates some patients' gums. Pretty much all of them seem to have this in it from what I can see, so if you get gum issues like sores after toothpaste exposure you might have to look deeper perhaps.
Essentially I am told that Stannous fluoride preparations in addition reducing decay like every fluoride toothpaste also helps desensitize teeth much more than sodium fluoride, and it has some anti-microbial properties reducing plaque, gingivitis, and significantly freshens breath.
He said before Stannous Fluoride, you were basically stuck getting products with Triclosan as an antimicrobial or potassium-nitrate to reduce sensitivity. Now you get all the benefits in one.
Pro-Tip: Spit don't rinse to keep the flouride in contact with your teeth for a while after brushing.
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I am sure there are more, but often it seems the big name brands make all sorts of different labels, packaging, and flavors for the same products. At this moment in time, these are the ones I found that do not mention whitening and have Stannous Fluoride:
Sensodyne Rapid Relief Mint
Sensodyne Rapid Relief Extra Fresh
Sensodyne Repair and Protect
Sensodyne Repair and Protect Extra Fresh
Sensodyne Complete Protection
Sensodyne Complete Protection Extra Fresh
Colgate Total SF Whole Mouth Health Clean Mint
Crest Gum Detoxify Deep Clean
Crest Pro-Health Sensitive and Enamel Shield
Crest Gum and Enamel Repair Toothpaste Intensive Clean
Crest Pro-Health Sensitivity Complete Protection
Crest Pro-Health Advanced Deep Clean Mint