The first thing I would do is power down the laptop; flip it over; remove all the obvious covers; and vacuum it out. You *could* blow it out, but I don't recommend that with enclosed systems like laptops, PSUs, etc. It sounds to me like you have a slow heat issue (assuming it is NOT what others have suggested with Malware and a bad memory card and such) and the CPU is overheating a little, which is causing the freeze after 30 minutes. It's not getting so hot that it just shuts down, which probably means the CPU fan is still spinning but it's having trouble moving the hot air away from the CPU. I'm willing to bet you'll find a lot of dust under those covers. Just be careful when you're vacuuming near the fans, because they're typically not very well attached and flop around a little...and getting them back in the right spots can be a bitch.
It sounds like you know how much memory you've installed, so check it by watching the first startup screen; it will display the amount of memory the system recognizes. (If it goes by too fast, use the Pause/Break key to freeze the screen.) Or, you can download this free app I've been using lately; it's amazing. It tells you everything that is in your computer, hardware-wise, down to the clock speeds and channel widths of your memory. So you can easily tell how much memory the system recognizes using the app, and then you'll know if you have a bad memory card. (The app is called CPU-Z; just Google it.)
However, if your system is displaying less memory installed than you expect to see, do not automatically assume the memory is bad; just try reseating it first. Take it out and put it back in. All of them.
In terms of buying a new desktop, I've been using a few different systems depending on what my sessions call for...I won't go into the details, but it sounds to me like you *might* be able to use what I refer to as my "small, quiet system" (as opposed to my "big, loud, dorky system with liquid cooling and enough power to compute cosmology modeling..." hahaha). It's based around the newer "mini-PCs" Intel has been selling. They refer to them as NUCs, or "Next Unit of Computing." To get the best bang for yer buck, they *do* require a bit of knowledge and assembly but it's much easier than assembling a new tower from the ground up. The first thing to consider is how many dedicated USB and SATA (external drive) connections you'll need, as these boxes are very small and don't have many connections. I can only use my NUC when I'm doing basic tracking (max 8, preferably less, especially at higher resolutions) and don't need USB connections for keyboard/surface controllers, as their dedicated CC ccommands won't work through USB hubs. Most NUCs have just enough USB connections for a keyboard and a mouse, with one or two leftover.
So, if you're just a solo/project studio kinda guy, a NUC might be the way to go. But like I said, they're sorta DIY, like a barebones kit, and if you want to get the most outta yer money, you'll want to source the parts yerself, but that means you need to have a basic working knowledge of how memory works in tandem with the front-side bus and the CPU and all that jazz. You can pay a little more and buy one preassembled too, but you'll pay 30% or more for the convenience. But...like I keep saying, depending on how much power you need, you can get an Intel NUC from Fry's right now, ready to go outta the box, for less than $170. And it fits in the palm of yer hand. Keep in mind, that price includes no peripherals: no monitor, no keyboard, no mouse. And it's a pretty weak system. Just using it to demonstrate the pricing on these things.
Hope that helps.