The head is important in two ways - the physical and the mental:

Physically, I see an awful lot of people with their head too far forward, or looking downwards, or whatever else. If you are having difficulty balancing and you've checked out your posture and shifting your weight, then it may well be down to your head being too far forward or leaning sideways. If you really can't move to straighten your neck, then it's time to see a chiropracter - your neck is really that important, not just for posture and balance, but also as it contains so many nerve connections, that may well affect your body in other ways too.

If you can move to straighten your neck, then practise doing so, by feeling the top of your head pushing upwards, while also tucking your chin. Do this in front of a mirror, until you know what it feels like, then check in with that feeling every so often throughout the day, so you can catch yourself and change your habitual "mad professor" posture into feeling much more relaxed.

Ok, onto mentally - the thing that will defeat you in tai chi, is your own thoughts. I myself was a complete klutz, a couch potato, with no physical skills, since I had always been useless at sport as a kid and therefore avoided it it later life too. From such an unpromising start, I have moved on to taking up tai chi, moving through its  levels, becoming an instructor. At every point, the hardest thing was overcoming my own ideas of my own lacks of physical prowess. Having a lack of physical prowess wasn't a barrier - just practising tai chi will improve your physicality - but the mental thoughts are another matter.

So many people I come across get discouraged because they aren't perfect from lesson one. I don't know where this is coming from, but the correct attitude is to welcome mistakes, to realise that making mistakes is how you make progress. Without mistakes, you can't understand how to improve. No-one has to be perfect, tai chi is a path, not an end. The only thing that really matters in tai chi, is perseverence. Body type is irrelevant, physical perfection is irrelevant, athleticism is irrelevant - persistence is key.

In my class, when I started, were a number of people who could all balance, who could remember things, who could pick up things quickly, who could move their feet correctly. Some of them stayed with tai chi, many of them gave up. The ones who gave up are gone and I am still here and improving. The ones who stayed all improved and I was well behind them for ages.... until  my last grading, when we all graded for the same level together. By persistence, I had caught up with them and reached the same level, despite my total lack of co-ordination, talent, or whatever, at the start.

I could have given up, but I just enjoyed doing tai chi, even when I was no good at it, or at anything physical, to start with. Not worrying about not being perfect - since I was so untalented compared to everyone else I saw, I had no need to worry about trying to equal them; it was't going to happen - I just got on with doing my own small progresses and over time these added up to major progress.

So, don't worry about not being perfect, perfection is irrelevant! Just keep doing your tai chi, enjoy it for itself, and progress will happen!