You probably already solved this, but just in case check all the wiring ends. The alternator does not affect starting much, except indirectly by not charging the battery properly. This is unlikely as you got a new battery and the issue persisted.
My 5.9 was starting weak, so I check the battery and it was bad. Replaced it and had stronger starts. A few weeks later, it weakened to the point of not turning over but the battery checked out fine and was charging properly. I cleaned the terminals at the starter and it started very strong for a while more, but weakened eventually to the point of not turning over all the time even though I could hear the relay (not the solenoid) click. I took apart my battery cables and didn't have enough slack to cut and expose fresh copper so I did something else. Today, I installed some marine terminals and crimped, then soldered all the wiring to some copper lugs. It's never started as strong as it does now. Looks like I was having problems at both ends of the cables and now all that I need to do is touch up the chassis ground since it's the last thing left to mess with.
First rule of electrical issues is to check, and re-check, then verify, and re-verify that adequate power is indeed getting to the circuit in question. Check all your wiring ends, then fix or replace if needed. It's cheaper than a starter and I for one am glad I haven't given up on mine, yet.
WARNING: Lo barato sale caro (Cheap things often lead to more expense)