Hi Bob,
Like so many things in SEO, I'd be looking at this from a UX point of view first and foremost, then do a quick revision through your SEO glasses before implementing.
Will making this H1 change help or hinder your user experience?
To offer my own opinion in response to this question, I think the answer is 'hinder'. As a user I would much rather see just the name of the shoe both on the category page and in the H1 of the product page.
It's clean, succinct and doesn't feel like someone had an SEO agenda when they crafted it. I'd even go so far as to say I'd think twice before handing over my credit card details to a site that used product titles like "Adidas Action Series Shoes size 5 through 10 for running, walking, and hiking" rather than simply "Adidas Action Series". Even the word "shoes" is quite redundant here in my opinion.
To give you an idea of a website that ranks very well in this field in Australia, check out HypeDC. They're not our client, nor have they ever been but they deserve that link because they do a great job in most areas and they highlight my point. They sell shoes (and all things shoe-like... sneakers, boots etc) and yet they don't jam the word(s) into every title, heading and paragraph they can.
If you're looking at the Nike Air Huarache Ultra shoe, that shoe's product title on the category page is simply "Nike Air Huarache Ultra". That's it. No mention of shoe, shoes, running, walking, hiking, footwear etc. The page title is almost identical and even the content on the product page has no mention of the word shoe!
While I wouldn't necessarily recommend avoiding the keyword like they have, it's a clear demonstration that Google understands your website as a whole so there's no need to go jamming all those keywords everywhere. Craft a pleasant user experience and make sure your website in clear, as a single unit, what you do.
You may be wondering how the hell HypeDC ranks so well without using the keyword even once? All I can offer is external suggestions but I'd say it's a combination of the following:
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They don't sell 'cheap shoes' so they know their target audience are "shoe people" who are pretty likely to know what they want already so I'd expect the search volume of specific shoe models is quite high. Just referencing the specific model on a product page likely works in their favor for these searches.
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Their link profile is highly relevant, as is their anchor profile. This means from the link profile alone Google can start to understand that HypeDC = shoe store.
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They have no shortage of pages on the site that relate directly to things in the shoe industry. A "shoe size guide", a bunch of shoe questions in the FAQ, a page for each of their shoe manufacturers etc.
I'm sure there are plenty more but this post is already getting long (sorry!) but you get the picture. Don't get caught up in "where and how exactly should I place my keyword in this element to appease the Google Gods?", just make a user friendly website and 9 times out of 10 you will have made a well-optimised one by accident!
I realise shoes is possibly not even your vertical but I'm just working with the example at hand. Everything I've just said can be applied to any industry because the core principles don't change.
I hope this helps!

