Seems like a reasonable assumption.
I got the parts for my 2 canister (one part, one carbon) whole house setup from a plumbing supply house for about $200 and spent an afternoon with a few handy friends installing it. It is far superior to the shower filter, which I don't use any more. With water as terrible as you're describing, it would probably be an excellent investment in your health.
If you can't install a whole house filter, I'd recommend a shower filter with 2 separate filters, one particulate and one carbon. I got this one for myself and it helped for quite a while until I got a bigger whole house system installed.
Just about any filter would probably help. I needed a special one because of all the particulates in my water, so I could change the particulate filter when needed without having to change the more expensive carbon filter.
They all attach between the shower head and the pipe, so there's no need for a different head. The basic kinds are ones that you have to unscrew the whole thing to change the filter, and ones that just kinda plug in from the side so you change the filter without removing the head. Mine is the second kind and I think it's probably much easier to deal with.
https://www.amazon.com/dp/B07JGMGP8W/
I have since installed a whole house filter that has 2 canisters, one for sediment and one carbon, and I'm finding far more effective. The parts cost about $150 at a plumbing supply shop and an afternoon with some handy friends to install. I do realize that this isn't an option for everyone, and the shower filter was a real blessing.
I actually do sebum only these days, so my brushing session is designed to clean my hair to what I consider clean. It usually takes about an hour. I also have curls and normal porosity hair so that greatly affects the results I need and get. I consider my hair clean when my scalp has had its weekly buildup of shed skin lifted, the buildup and other environmental debris has been removed by combing, and my oils have been refreshed and spread throughout my hair. Then I apply my weekly moisture treatment and rinse it off later and set my curls. I don't actually use anything wet to wash my hair anymore, lol. I also need oils in my hair to give my curls some structure and support and seal in their moisture. Curly hair doesn't like being too 'clean'.
You'll need to experiment with different techniques, like jenifarr said, and find what works for you and your hair. But it honestly sounds like you're almost there. As I said before, I think you just need some dry mechanical cleaning and figuring out how to most effectively use it with your specific hair and needs.
If you need to, once you've given it a try and have an idea of how it's working, come back and we would be happy to give advice on specific techniques to help you reach your goals.