I've had bad posture all my life. I recently have been practicing standing straight but when I do my lower spine hurts. Is this normal?

It may be if your . Core musculature is weak as you are not accustomed to stand in a balanced manner. Regular workouts incorporating stretching, cardio and core muscle strengthening like planks or even yoga or pilates will help.
Posture and Pain. You are doing yourself a great favor by bringing more attention to your posture. Unfortunately, most people tend to tense up one region or another when trying to correct posture. Investigate something called alexander technique further and find a teacher if one is available. It has been studied in medical journals, and is an elegant method of addressing complex problems associated with back pain.