It doesn’t matter where you start. Just start.
It's a well known issue for years with the HCx, well known to Garmin too. I found that changing the mount helped (go for a firm rather than a spongy mount). It doesn't always help. I wouldn't make a complaint to the UK centre, just email them normally and see what they suggest. I have always found them to be extremely helpful.
It also had the constant shutoff problems, but I thought this was normal.
I found the switching off problem is easily fixed with a bit of tape on the bike mount. ie a couple of layers of duct tape inside the bit where the clip fits. This gives a nice snug fit, which reduces vibration.
The problem is endemic to all Garmins that use 2xAAs (interesting that the 705 does it too) - even the newest models - but later models have a much more secure bike mount so the problem is much less noticeable.