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I've mostly gotten the quality to where I'm happy except for whatever it's doing to the bow of Benchy. Still some stringing but I don't expect that to go away completely with TPU. Hatchbox TPU, been lowering temp, lowering print speed, fine tuning retraction, etc. I can give exact numbers if needed, but if anyone knows what to adjust specifically for what it's doing at the bow that would be great.
Additional pictures: 
What printer and what orientation did you print these?
Almost looks like some of the ones I've done in abs where I didn't get proper cooling (turned to have the bow @ 45°, 90° being straight toward the rear of the printer for reference) either from my fan settings or orientation, could try increasing the fan speed? I know you said you've dropped the temperature but personally I still find I need to have the part cooling fan on (do print enclosed though so YMMV)
How dry is the filament, afaik tpu is hygroscopic (been a while since I've printed with it unfortunately), I've seen messy bow show up on less dry filament, always worth a try anyhow just to remove variables.
Prusa MK4S, standard Benchy orientation. I haven't tried adjusting cooling yet, just using the default FLEX settings but I can try that.
Could try turning the bow forward and see if that changes anything. From what I've read, start low with the fan, like 10-15%.
All else fails, still say worth drying it if you can, I've had some really damp spools fresh out of the bag before.
It was from the bag and been printing in my enclosure, so that alone tends to dry out my filament but it is my first time with TPU and I know it soaks up moisture. I've never had issues with wet filament, would it cause an issue in 1 area like that?
It'll cause more zits and the like, more stringing, don't know that'd 100% cause the issue but certainly won't help.
I just did a round of nylon last week which is also super hygroscopic, bag's like at most 8 months old but seal was intact. Even printing out of a drybox with fresh desiccant I noticed more stringing and occasional blobs + nozzle buildup over the few days (Was redoing my hotend so I inspected it, no sign of nozzle leaking), can definitely make overhangs worse.
Supports? I'm guessing no, and Ive had issues with TPU and overhangs like the bow there myself. TPU is flexible and thus sags more when printing overhangs.
No supports, the Internet (and if you can't trust the Internet who CAN you trust?) said I shouldn't need it for a Benchy.
I don't for PLA/PLA+, but TPU is different. At least in my experience.