Handle replacement with keyed handle
Hello there. Nope, you should be fine just replacing the handle set. See our 82-030A or 82-030B. These are both keyed handle sets, but they differ in the position of the inside thumb lever and the outdoor key lock. The 82-030A has both the thumb lever and key lock perfectly centered between the mounting holes. The 82-030B has them offset towards the lower mounting holes. Judging from your photo I think your thumb lever is offset, and so you'd need the 82-030B. But photos can be tricky to judge, so double-check this just to be safe. Let me know if you have further questions.
This can sometimes happen when you attempt to replace a non-keyed handle set with one that's keyed. Essentially, your door was not designed to accept a key lock, so our solution is to make it accept it.
To do this, you will need to take a rat tail file and carefully widen the hole that takes the key cylinder. There's no trick to this aside from patience. It takes a little time and trial and error, but it will fix this issue entirely.