Jeremy,
I would venture to say that most folded and welded axes show the weld near the eye. The possibility of burning the eye means you have to soak the piece and allow the thick area that is to be welded to come up to welding heat slowly. Never the less, it will leave some lines showing most of the time. Just make sure that it is welded good. I do this by bringing it up to welding heat several times and bumping it but not reducing the thickness much until I'm satisfied that it is truly welded. Then on the last welding heat, hammer it to thickness.
Here is the top and bottom of that hawk.
This is what I do on some. I will make the forge weld and while it is red hot I will slap it in the vise and arc weld the inside of the eye with one bead being careful to stay in the boundaries of the top and bottom rim. While I believe it will hold for the drifting without the arc weld, I don't want to chance that at this point. It is just insurance that what I have gained I can keep. I really don't care if the forge weld lines show some. Most historic pieces showed after all. If I see the need to make one that is really dressed up, I may take more pains to close everything even if I have to fill a small gap with an arc weld. I usually can close it all up by hammering it but sometimes it just wont cooperate. That's the reality of it and I wont let that stop the project because of my pride.
Making these hatchets and hawks will take practice. Don't think you can just start making them super clean at first. I have learned a couple things as I have spent so much time making them and some of those things are hard to convey. One thing I can tell you. When you are stepping the bar down and back up for the eye walls, don't make the step 90 degrees. Make it about a 45 degree angle. When the two 45 degree angles meet in the front of the eye, that will guide the drift to the center getting things in line right at the start. That is important.