I've tried looking up why Gandalf claims to be Saruman but I keep finding things saying 'Gandalf is trying to show that he is as Saruman should have been - how he shouldn't be corrupt' is this correct?
The relevant quote is:
Yes, I am white now,' said Gandalf. 'Indeed I am Saruman, one might almost say, Saruman as he should have been.
When the Istari arrived in Middle-Earth, they were as emissaries from Valinor. They no longer wanted to interfere directly in the affairs of Middle-Earth, so the 'wizards' were sent to help the free peoples of middle-earth in their fight against Sauron. You can discern this intent by what Cirdan the Shipwright said when he gave the elven-ring Narya to Gandalf:
"Take this ring, master," he said, "for your labours will be heavy; but it will support you in the weariness that you have taken upon yourself. For this is the Ring of Fire, and with it you may rekindle hearts in a world that grows chill."
They were, however, forbidden to "dominate the peoples of Middle-Earth or match power with power". Saruman was set up as the leader of the order, and as such bore great responsibility in carrying out this task. Thus, when he accumulated power for himself and attempted to dominate the peoples of Middle-Earth, he was in direct opposition to his orders.
Therefore, Gandalf, when he was 'promoted' after his resurrection, cast Saruman from the order. He then took the lead in inspiring the free peoples fighting against Sauron. In this capacity, he was fulfilling the role Saruman was sent for, indeed "Saruman as he should have been."
In Christian theology, Adam represents all humans, their potential and their fall. (Adam literally means "Human" or "Earth"). As told by Romans 5, Christ is the new human. He is what Adam should have been.
Tolkien was deeply religious. I suspect he may have seen Gandalf as the foil of Saruman, as representing what Saruman could have become.