I am not in the third trimester but I didn't want to leave you with no answer
with pregnancy, the weight gain puts alot of pressure on our lower backs. The shooting pain that you are describing sounds like you may have a pinched sciatic nerve and when you cough you irritate it.
I had this pre pregnancy and let me tell you it hurts like hell!! I almost collapse because it feels like a lightening bolt of pain down you leg!
I would recommend doing some back stretches and pelvic tilts.
Pelvic tilts- stand up with your back against the wall and tilt your pelvis back against the wall (basically pushing your lower back towards the wall. Do that 2x a day about 10-15 reps.
Another thing that helps me is a yoga back stretch called bird-dog. Google for an explanation. Its hard for me to describe ha.
Also try and ice your lower back to help with any swelling around the nerve.
Something I have been doing that also helps is to see a chiropractor. Alot of them are accustomed to working with pregnant patients. And it helps increase blood flow to you and baby and relieves pressure in your back.
Sorry to drop so much info on you! I hope this helps!