A couple of weeks ago, I attempted my first two grafts, grafting some UCD Calvert cuttings onto two branches of my Texas Everbearing. 

Well...I'm starting to think that I may not have been successful because I'm starting to see buds breaking on the TE just below the graft point.

Both grafts (Calverts) are still green, so they don't appear to have dried up (yet?).

Is this normal, or does it sound like I should re-attempt the graft?  If so, assuming the Calverts are still green, can I just cut the ends of both the TE and the Calvert and re-attempt grafting the same cuttings to the same branch???

If it's normal, should I try to remove the buds forming on the TE to get the auxins to try to move to the graft?

Any advice (hopefully constructive and experienced preferred, of course!) is welcome!

Thanks!

I'd definitely cut off the nearby growing TE buds.  If you lined up the green cambium between host and scion and kept it that way while binding them together I wouldn't touch it.  Without knowing what you did it's hard to give any further advice.

I agree to remove most/all of the buds for several inches up-branch from your graft, an attempt to have the rootstock use its energy on the graft. 

While I usually see the graft's buds growing within a couple of weeks of grafting, some have taken longer.  I'd wait, until failure is clear. 

If failed, if you wish to try again, cut the rootstock just up-branch from the failed graft and use a new scion cutting.  If the current graft has not taken, the cutting is not likely to be adequately viable for retrying/grafting after being out in the sun for several weeks, even wrapped in parafilm.

What kind of graft did you use?  I try to do 2or more of each scion variety for insurance: a graft on the end of the rootstock branch, and 1-2 chip grafts further up-branch.

Pics?

Thanks guys!  I'll try to post some updated pics tomorrow.  I did a whip & tongue graft on both and they seemed pretty solid when I wrapped them, but, I also know my dog is always chasing lizards around the planters and I know he brushes into the branches, so my biggest concern is that he somehow dislodged the graft while lizard hunting, even though it still feels in place THROUGH the wrappings.

I didn't document the grafting process, but here are a couple of pics I had posted after I had first performed the grafts:

 

 

Buds are breaking on the first couple of nodes below the wraps on each graft.  On a positive note, both branches still have the nice green color, so they don't appear to be dead yet.

I'll wait a while to see what happens since it's only been two weeks, but I'm somewhat less hopeful than I was immediately after performing what I thought was a great grafting operation! ;)

For sure signs is the drying and wrinkling of the cutting....so it may take a few weeks more....what I do, to check, is to get a thin razor and make a thin slice, if it took, the milk will start to drip out....make sure you cut around the middle not towards the top...(the top may have dried up)