Like others have said, they can pattern differently on different ranches and could be hard-horned anytime during the year. It is my experience and observation that herds tend to "syncronize" so far as their horn drops. For example, if you have a ranch with 1 herd and they drop in Feb - then most of the herd will drop in Feb. Multiple herds on a ranch, and each can follow their own pattern. Of course, there are always a couple that want to be different.
That being said, MOST of the axis statewide in Texas are hard-horned by May - even the free range animals. And the peak of the rut seems to happen mid July.
Personally, I do not like hunting the rut for axis. I find they are much harder to pattern, unless you are a relatively small HF ranch (under 400) acres. Hunting gets really tough for axis in August.
Summer months pattern on water. They will drink at least once a day, and if not rutting, will have stable and predictable paths from/to water.
Free range animals tend to be harder to pattern, or at least - their pattern may span days to weeks, rather than a daily pattern you typically see on HF ranches.
For me, the best month to hunt axis seems to be Feb. By Feb, they are hitting feeders hard and predictable, their coats are thick and fluffy, and they are their "beefiest". However, it is harder to find fully developed hard-horned animals at this time. It is definitely the best time for does.
Velvet Axis are best in late March to early May, the later the better horns. Obviously you want to catch them just before they rub out.
June and July are good months to get a look at most horns, and by sheer numbers, probably the best for largest horns. But again, for me personally - I see a lot of horns in June/July, but those 35"+ critters I see just as many in Jan/Feb - or in the fall - in the free range areas that I hunt (that's talking 2-3 animals that size). Defintitely lots more animals in the 26-32" size in the summer.
Summer hunting can be good, but it can be HOT. Box blinds in the summer can reach 120+ degrees and can make hunting miserable - you sweat, you stink, and animals bust you easily. In the heat of the summer, on both HF and free range, there seems to be movement in the morning/evening right at daybreak as they travel to water. The rest of the time they are WAY deep in heavy cover. While it is true that axis have a annoying habit of showing up anytime occaisionally during the day - sitting long summer days from 6am to 9pm in 120 deg blind is tough on the hunter. And it always seems when you take a mid day break, they show up.
Your .270 is plenty good caliber - but be prepared for little or no bleeding. I would NOT use a Barnes TTSX bullet on axis for this reason - they passthru leaving just small holes. Anyone who has shot a number of axis will tell you that their skin seems to slide over bullet holes and it is common to have little or no blood trail, even with a good shot. They are TOUGH animals too! I've had bucks take 3 shots into the heart/lung area before they would stay down. And then again, I've had them drop right over. My daughter took at 35" axis with a .270 Fusion this past May and he ran 20 yards without a drop of blood, jump a fence, then just drop cold.
I prefer not to hunt the full moon.
Good luck and enjoy. Axis deer are my passion.