Greetings, welcome to the Forums.
Yes indeed, many cactus, including your White Bunny-ear Opuntia (Opuntia microdasys 'Albispina') will fold like an accordion when drought-stressed and will often recover fully when water is again received. However, even a cactus can dry out to the point of death, and furthermore, rot can sometimes masquerade as normal dehydration. In temperate climes, it is often wise to not water cactus at all in the colder Winter months,t though if temperatures and light are high enough this may be done. Try to give your cactus as much sunlight as possible in Winter, and throughout the year. as always, fast-draining soil that mostly dries between irrigation episodes is the first imperative.
By all means do remove a pad and start a separate plant. Holding the pad with gloves and tongs or thick cardboard during the procedure is essential to avoid a minor but annoying injury to yourself.
