I've just installed a very new, very high powered graphics card on an Intel i5 750 system, and it performed poorly until I overclocked it by adjusting the base clock, from 2.6GHz to 3.30GHz (base clock from 133 -> 160-170). Did not make any other adjustments (e.g. to voltage, etc) and it was stable but overheating, so I bought an aftermarket pre-sealed CPU water cooler ($50!) and installed it to keep the temp stable. I also followed some guides on tweaking the OS, e.g. setting 'high power' mode, using mobo tweaking tools, etc. Games are now very playable, and brought proc performance in-line with my requirements, without the purchase of a new computer. This chip is famously overclockable, so before you junk it, try overclocking it! If it works, don't allow the chip to overheat too much. Quickly get one of these cooling systems put in and keep going (ideally, get it installed by someone who knows what they're doing). If it doesn't satisfy, you can simply put the cooling system you purchased in with your new cpu/mobo. Extend the life of your PC. Try to get the best ROI from your current and next machine by going as long as possible between machines. New hardware (e.g. thunderbolt 3, usb 3.1 type c connectors, NVMe, 10nm chips, faster RAM, bigger SSDs, etc) is always just around the corner ... Just buy the minimal to keep your machine doing what you need, and think long term for each part... e.g. if you buy an SSD now, you can use that in a new machine eventually... so why not just buy that part now? Keep iterating until you have enough performance, don't do a major upgrade unless you're sure it's completely justified.
... BTW your computer sounds like it could easily take a new cooling solution. If its in a standard PC case and has a standard MSI motherboard, I doubt it'll be a problem.