If you are new to playing, or don't play that often and haven't developed hardened fingertips, this echo you hear is likely the string sticking to your finger tips like it does with mine from time to time.

If the skin on your fingertips is soft, lkie mine usually is, it will wrap around the string when you fret and actually adhere slightly...hence causing you to inadvertently "pluck" the string when you change strings. This causes that unwanted and annoying resonance. The only suggestion I can make from experience is to play and toughen up your fingertips. Make sure your fingers are clean and dry as well. I find personally that damp/dirty fingers/finger tips are softer and tend to stick more.

Of course this is just my two cents based on my limited experience and observations when I am practicing. Hope something in this rambling is helpful :)

All the best!

G.