Need help with oil options

Ok I'll start off with what's going on, I put a new cam and lifters in my 340 last year, I broke the motor in with Lucas oil zinc additive and 30w rotella, I drove the car maybe 500 miles, had a couple issues and come to find out the rings were all bad. I built a fresh 360, just had the crank, rods, pistons balanced, new rings and all new bearings on the bottom end, I reused my top end because it was fine. Now I know I don't need to break the cam in but I have all the other bearings and rings to break in. My issue is do I use the same oil and additive I did before and run it with zinc and rotella? Or I was wanting to switch to like royal purple synthetic that has extra zinc in the oil, or maybe another brand. If I switch oil types should I break it in differently? Or should I not go synthetic? This car is a street car prob won't see track, long cruises and have some fun. Opinions please