Nothing wrong with another string ringing out if that is your intention - but you should also be able to a) play a note for its intended length and, b) play clean notes.
When you shift notes on the same string you do this automatically - when you shift strings you need to mute. For bigger than a single string jump down, you generally use the palm of your picking hand - for a bigger than single string jump up, you can often use the underside of your finger or your palm. On adjacent strings using the rolling technique is not only efficient in terms of speed, it also gives you another option for muting.
If you practice all forms of muting, you just tend to automatically use the most efficient method when playing. The unintended ringing strings aren't generally as noticeable on an unplugged acoustic, but plug it in, or crank the volume up on an electric guitar and they can sound dreadful.
As for your barres - all the strings will be notes from the chord you are playing (even the ones forming the barre) - you may want them all to ring out - very different to playing single notes. If you don't want all of the strings to ring out on barres, you can add pick control to one of your muting techniques.
Welcome to the forum