iPhone 5s display behaves strangly on swipe/click
This sounds like the digitizer is failing.

The digitizer is a "glass layer" that sits on top of your LCD display to accept input (taps, swipes, etc.).  This can begin to fail for any number of reasons from physical damage, to water damage or even aging.

The iPhone 5S has the digitizer and LCD as a single assembly, so to replace one, you'll have to replace both.  iFixit has an excellent guide on how to do this.  However, if doing this type of procedure is not your cup of tea, you should take it in.

(Though the 5S is 6 years old now; it's probably time for a new phone).