What do you call a person who avoids their responsibilities by disregarding them and just continue doing other things?
To shirk is to avoid or neglect a responsibility.
So, a person who shirks is a shirker.
Slacker might be a good choice: a person who shirks work or obligation; especially : one who evades military service in time of war (Merriam-Webster)
Procrastinate: to be slow or late about doing something that should be done : to delay doing something until a later time because you do not want to do it, because you are lazy, etc. (Merriam-Webster)