I speak from a Lutheran perspective. In our confession we say this - AC II.1. 1] Also they teach that since the fall of Adam all men begotten in the natural way are born with sin, that is, without the fear of God, without trust in God, and with 2] concupiscence; and that this disease, or vice of origin, is truly sin, even now condemning and bringing eternal death upon those not born again through Baptism and the Holy Ghost.
In general I think we would object to the terminology of TD as it is problematic. The reason is that it can mean that man is not able to do what is ethically good or from a human point of view. We know there are atheists who,from the outside, appear to be ethical, even Jesus says that we as parents though evil do good to our children, so one looking at TD will find TD puzzling. This is not what Original Sin means. What we mean is that when it comes to moving towards God, having faith in Him, trusting Him or fearing Him and doing what he commands we are not able to do this on our own since we are impure and are not able to do what his Laws demand.
In Luther's Small Catechism - we have this...I believe that I cannot by my own reason or strength believe in Jesus Christ, my Lord, or come to Him; but the Holy Ghost has called me by the Gospel, enlightened me with His gifts, sanctified and kept me in the true faith; even as He calls, gathers, enlightens, and sanctifies the whole Christian Church on earth, and keeps it with Jesus Christ in the one true faith; in which Christian Church He forgives daily and richly all sins to me and all believers, and at the last day will raise up me and all the dead, and will give to me and to all believers in Christ everlasting life. This is most certainly true.
Also yes that is correct we would rather talk about Original Sin rather than get to the paradigm and terminology of TD.