Resurectiom or Reincarnation

Many years ago–more than forty now–we had a gathering in Durham NC, where we invited the Bishop of Durham to speak. In the Q&A period someone asked him about reincarnation and resurrection and about Paul in his First Letter to the Corinthians seeming to give a view of resurrection diametrically opposed to that reported in the gospels as given by Jesus.
Only many years later did I look up the references to the two topics. I found that the Bible seemed to be saying two different–opposite–things. In Corinthians it is obvious that Paul is trying to substantiate a belief based on the total regeneration of the physical body. In fact he said “…[H]ow can some of you say there is no resurrection of the dead? If there be no resurrection, then Christ was not raised; and if Christ was not raised, then our gospel is null and void, and so is your faith. But the truth is, Christ was raised to life–the firstfruits of the harvest of the dead. For since it was a man who brought death into the world, a man also brought resurrection of the dead.” [I Cor XV:13-15] *
In Matthew’s version of the gospel, we find Jesus answering a question about resurrection: ‘Master, Moses said, “If a man should die childless, his brother shall marry the widow and carry on his brother’s family.” Now we knew of seven brothers. The first married and died, and as he was without issue his wife was left to his brother. The same thing happened with the second, and the third, and so on with all seven. Last of all the woman died. At the resurrection, then, whose wife will she be, for they had all married her?’
Jesus answered, ‘You are mistaken, because you know neither the scriptures nor the power of God. At the resurrection men and women do not marry; they are like angels in heaven.’ [Matthew XXII:24] *
When thinking about reincarnation, we think about spiritual resurrection, though we are told the spirit actually proceeds to inhabit a new and different body.
It seems to us that if the Christians would take the next step in the teachings attributed to Jesus (that is, that angelic spirits can now come to earth), Christianity would align itself with all the major eastern religions–and it could have no argument with Wicca. This begs the question, though, of actual raising of the “dead”.
The miracles of Jesus come to us purely as hearsay. There is no hard evidence. Surely it’s not possible that a group of people trying to establish themselves as a new religion would actually fake it?
The question of Lazarus is somewhat different: Supposedly there were many witnesses. We also have the myth of Sir Lancelot del Oc reanimating a knight whom he had just killed: a myth based on a myth.

* All quotes from The New English Bible Oxford University Press 1970