Open your text message app. Click the 3 dots in the upper right hand corner. Click advanced messages and turn it off. Itll take away the dot on contacts icon. Before turning it off. Itll notify of the benefits and cons of doing so.
The little blue dot is an indicator for RCS features. These are an advance messaging service across android devices. Helps you share larger files. Can send photos and videos up to 100mb.
AT&T explains it more here:https://www.att.com/features/advanced-messaging/
hope this helps.