@EmpressShadow7 Since I don't know where you live, I can't possibly tell you what's available and cheaper than an Omen. But this list is a good place to start:
https://www.notebookcheck.net/The-Best-Budget-Gaming-Notebooks.262742.0.html
There are of course a few caveats. One is that "budget" means different things to different people, not to mention that prices can very significantly by region. Another is that not all of these run cool, and some of the ones that do are somewhat loud. You can skip all the technical info and scroll down to the overall ratings towards the bottom: you'll notice that, for example, the Lenovo Legion runs cool and quiet, the Dell is cool and the MSI quiet, and the Aorus is neither. These are relative rankings of course, but anything above 90% is excellent, and you'd probably be fine with anything above 85.
Please also be aware that the same brand name can run differently depending on the processor it has. These days, laptops with AMD CPUs generally run cooler than ones with Intel CPUs because the former is more power efficient than the latter. There are exceptions, e.g. the Acer Predator Helios, but you'll notice it's apparently quite loud under load to compensate. The Lenovo Legion has the advantage, cooling-wise of a weaker processor, although it's still perfectly fine.
I'd suggest the list more as a starting point than anything else: you can compare its offerings with what's available in your region and go from there. But of course it's best to look up reviews of the specific option(s) you're considering before making a purchase. I can help with that too, if you want to list a few models.
And again, you don't need to pay for a 2070; a 3060 is just as fast and is probably going to be cheaper if you can find one available.