No, in fact, Catholics don't believe in a "sin nature". That's a term Protestants use. We believe that Adam lost the ability to transmit the divine indwelling life of God to his descendants (through natural means) in the Fall. You don't have to "transmit" the absence of something.
Sin is simply missing the mark of God's Plan. Whereas in God's Plan, man was originally created with such an ability, the loss of that ability and the natural condemnation of all man following from Adam genetically, results what is called an Old sin nature, or the natural man, as discerned from the new man or being reborn in the spirit.
The old sin nature is very much a Catholic concept. What might be more associated with many somewhat sloppy, but popular 20th century Protestant denominations term the "old sin nature", are associations of temptation and evil thinking with the "old sin nature", misidentifying it with demonic influence or antiChristian thinking. Old Sin Nature (OSN) has more to do with mental attitude sins with the human thinking them being solely responsible for their origination, not another spiritual person placing those thoughts in our minds. We can be saved, in fellowship with God, advancing in sanctification by His work in us, when out of the blue, independent of any other person, in our thinking we are distracted or tempted by our OSN, our scarred way of thinking, our natural way of appealing to good and evil as a substitute for faith alone in Christ alone and what God has provided for us, and slip first in a mental attitude independent from God's Plan, ergo fall out of fellowship with Him, without performing any act or thought of evil.