Welcome to Android Central.
I don't have Textra but I understand your question, which raises a few questions.
First it always helps if we know what device you have. Not always but often advice can change based on the features of the device.
How are you accessing the messages? Tapping on them from the notification bar or opening the app?
I've found that unread/notification badge counts will remain if there is a notification in the notification bar. So if I get a notification from, could be any app really but I'll use Tapatalk as an example, Tapatalk and I open the app and read through my subscriptions. When I close the app the badge doesn't clear. It doesn't clear because the notifications remain in the notification bar. On the other hand if I tap the notification and have that open the app the badge and count are automatically updated. In your first line this seems to match what you're talking about if you're just opening the app to view these messages.
In your last lines, having the ability to mark threads read or unread is a feature of the app and would be something you would have to bring up to the app developers.
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