I got some weird dry skin/dandruff/oily scalp issue combo. Things that I have found working over a very long time struggling:

• Out of all shampoos Head and shoulders actually has the best effect. Anything else not-specialized doesn’t. Ketozonol/Nizoral does help but I am not feeling much better over HnS. And these are not suppsoed to be used long term.
• I have tried SLES shampoo from Neutrogena. Didnt work for me like at all.
• Water hardness is super important. My hometown had much softer water that allowed me to not to wash my head for a week and it would be fine. My current location has fucking insane hardness and I have to wash my head each 2-4 days or it starts to itch very badly. Whenever I travel, some of the time my hair feels great and I suspect that water hardness in the hotel might be much lower than at home. Maybe look into installing some kind of filter. If you live in a house, 100% invest in reverse osmosis filter at least for your bath/shower.
• My head also gets oily quick. I usually wash with 3 shampoos. 1st shampoo - cheap 1 euro shampoo that I use just to wash majority of oil off. 2nd shampoo - deep cleaning shampoo that is also cheap - 2.50eur but it does have an effect. 3rd is HnS. The longer I soak hair with 2nd and 3rd shampoos, the better it feels afterwards.
• Adding up to soaking point: if I take bath I soap my head quick, and each shampoo soaks my scalp for 20 minutes or so. This way head feels much nicer over the next days. In shower, 2-5 mitnues or so.
• adding up on deep cleaning shampoo: I found a few shampoos that claim they are deep cleaning/cleansing. But upon trying, only one felt effective. If you seek for these shampoos, try more options til you find something that works.
• Conditioners have no effect to me. I have heard that it might help in certain cases, so maybe you should try that too.

I really do hope you will find a solution to your problem. This shit sucks donkey balls and is so annoying to live with.