Thread is going in circles...
If you want to build a new PC post your budget, the resolution and refresh rate of your monitors, if they are old 1080p screens do you want new 1440p?
List the spec of your current PC in case anything can be re-used, unlikely but for all we know you have a modern GPU in an old PC.
Then a bunch of people will come along and post you some build lists; note it might be better to create a new thread for the PC build in the correct sub forum and post the link at the bottom of this one.
If you list what apps you use and if there’s an upcoming game you are really looking forward to people will have a better idea how to balance the build.
Re your current PC
I will contradict a few of the other posts in that my ancient PC with very little ram is quiet capable of having loads of browser tabs open without any problems. Yes if building a new PC a few years ago 16GB was the go to ram configuration and 32GB is the go to configuration now unless you have particular needs then more might be recommended but browsers can run with much less.
It’s possible there’s a hardware or software issue with your current PC because apps shouldn’t crash just because the CPU is under high load. Yes things can become slow and unresponsive if the CPU is pegged near 100% but that isn’t the same as them crashing.
You have been very vague as to what apps you are using, some websites are far more demanding than others and we don’t even know which browser you are using.
Two things I noticed recently that caused slow downs on my ancient W10 PC:
1. Microsoft update reinstalled CompatTelRunner.exe and it was scanning my system, I took ownership of it and deleted it again...
2. I like to try different browsers from time to time and Opera had this stupid animation of a cat as the background that caused massive CPU spikes until I changed the background to a static image.
Also an SSD is a must if your old PC has a mechanical HDD just swapping that makes a huge difference. You might also find a cheap second hand ram kit you could fit.
Re: 9800x3d or 9700x / more cores
The IPC is higher on modern processors, they also have higher clock speeds so a current gen 6 core CPU is already a lot faster than your old 6 core CPU.
The 3d chips give a boost in gaming but you pay a price premium for them sometimes its worth it other times its better to spend the money else where e.g. the GPU.
If it was all down to cores then everyone would be buying 64 core thread ripper processors but they are overkill for most users.
9800x3d = 8 core 5.2Ghz +3d cache @ £399.95
9700x = 8 core 5.5Ghz @ £269.99
They are very different chips if you wanted more cores because the software you are using is very multi threaded over the cache the 3d chips have then there’s the 9900x to consider with 12 cores which overclockers have for £349.99
9600x = 6 core 5.4Ghz @ £188.99 has around double the performance of your CPU in a lot of bench marks:
So starting with a lower end chip then upgrading it when AMD release their next batch of processors is another option.