Others have already hit a lot of good points. I appreciate your enthusiasm, but will be blunt. Your form makes it look like you have never used an axe. Get a hatchet and do something useful like chopping firewood for a while and then look at your form with a better understanding of the tool.
1. The axes you have are
not suited to thrusting.
2. Most of your strikes look ineffective. Even in your swinging movements, the blade is often out of line with the strike. If you hit somebody like that, you would, however, make them really angry.
3. You state that your motivation is to interpret a couple of traditional TKD empty handed forms for use with an axe. This is not necessarily a good idea if the movements in those forms do not involve a lot of hacking and slashing.
It is a cliche that "Weapons are an extension of the empty hands". Yes, there is a bit of truth to that, but you must at least consider what kinds of hand movements will translate to any given weapon, and even then, make considerable changes in your movements for them to be functional. For example, look at the following kama form. A kama shares some axe like qualities, and comes from a tradition (Karate) that, overall, shares more movements with TKD than say ...kung fu.
Now, I know nothing about traditional
Kobudo, but I do train with weapons (FMA and WC). When I look at this clip, I do not find it hard to imagine what the applications of these movements might be. Unfortunately, I cannot say the same for your form. Perhaps you should consult with someone with actual training in weapons.
Oh, one unrelated question. In the background I see a couple of Mook Yang Jong (WC wooden dummies). Does your TKD school teach those as well?