Yet more attempts at grafting apples!

If you're cutting limbs off to graft onto you should be doing it in winter when the tree is dormant, when the sap rises in the spring it will flow into the branch and either create a union or scab over the cut. What sort of diameter are you thinking of cutting. It may be easier to chip bud or splice the scion in before removing the original limb after the graft has taken, to ensure the sap still flows.