If your problems occur only while playing that one game, it is most likely the game itself causing them, not your hardware. You say you have tested some parts of your machine, but no details. Perhaps you can tell us more. See end of this post for my suggestions.
I can think of four ways this can happen, and the fix depends on the cause.
1. There actually is a hardware-related issue in the sense that something is not responding fast enough. That MIGHT be actually a driver issue. For that, you could try to update all the device drivers in the old machine in case that helps.
2. Maybe the machine has been set up with overclocking of the CPU or RAM, or with slightly low voltage to the CPU or RAM. These changes normally can be made in BIOS Setup. You could look through BIOS Setup to make sure things are set to normal specs for the components you have.
3. Maybe some of the files of the game are corrupted somehow. This could happen on this one machine, but not on others where you run it. A corrupted file does NOT mean the HDD is faulty, and it may NOT be detected by software looking for HDD flaws. It can happen simply if erroneous data is written in a perfectly normal manner. If you have original installation disks, try to uninstall the game completely, clean out the Recycle Bin, then reboot. Then re-install the the game, which will renew all its files.
4. I suppose it is possible, also, that in setting options in the game you have set something odd that this machine does not handle well.
If you want to do further testing of your hardware, I suggest the following:
5. For MEMORY, get MEMTEST 86+ and run it several times - maybe even overnight.
6. For the hard drive, some of the best, easiest and free testing utilities are from the maker of your HDD. So you need to know who made it. Then you go to their website and look for HDD diagnostic utilities to download. You'll have to pay attention to which form they are in. IF the HDD is by Western Digital, get their Data Lifeguard utility. If the HDD is from Seagate, get their Seatools. If from another maker, look around their website. Many of these have versions that run as applications under Windows. They also have versions that can run even when your can't get Windows to boot up. I prefer the latter, often designated as "for DOS" versions. Some of these are small enough you can write them to a single floppy diskette (IF you have a floppy drive!). Others come as an .iso file image of a CD. For an .iso file you must download the file, then use some CD burning software (like Nero) to burn the .iso image onto a CD-R disk. In either case, you then set your machine's Boot Sequence to boot from the medium (floppy or CD) you're using, put in the disk and boot up. The machine will boot from that disk, load a small version of DOS into its RAM, and run from there with no need for a fully functional HDD or other Operating System. You then get a menu of tests to run.
When using disk diagnostic suites, read the on-screen messages carefully. There usually are several tests you can run that are NOT destructive to your data and they are very useful. If you do get error messages, write them down and note them here. These suites also usually include more severe tests and repair tools that DO destroy data, so you must have your data backed up before using those particular tools. Such tools almost always warn you that they will destroy data and ask your permission to proceed. Very often you do NOT need these tools unless you have already determined with the NON-destructive tests that there are problems to be solved.
7. Windows' tool CHKDSK will check for faulty sectors of the HDD, and for garbled File System information. It cannot do all of the hardware testing that a HDD maker's diagnostics does. On the other hand, those diagnostics don't test for File System errors, so use both types of testing.