Try switching to VW/VH or % units for margins and padding, it can help to alleviate those issues, you can also try responsively.app to check various screen sizes from your PC.
I only needed browserstack to test Safari. Then i found this: https://schepp.dev/posts/running-webkit-on-windows/ for windows and the Gnome Web Browser for linux are sufficient.
Theres also https://responsively.app/ which is nice for screen sizes.
For automated testing, I guess Playwright can do the job, but I've not used it. https://playwright.dev/
There is also a tool called responsively app. https://responsively.app/
This will also show you how your pages display on each kind of device. It was super useful for me when building a portfolio
Check this out for simulating tons of devices at once. https://responsively.app/
I personally don't use rem or em much because the design and typesetting is the most important part of my projects.
>don't want to spend on buying multiple devices.
You do realise that chrome dev tools has the option to select a device to simulate screen size and speed.
Apart from it, I contributed to a tool/app responsively during the summer which solves the responsive part of your problem.
I can't comment on the browser compatibility part.