If you've replaced RAM, you can replace a CPU.
Just carefully remove the fan over top the CPU housing, and you'll find the chip. Destatic your hand, and then slowly pull it out straight (Some models of motherboard have a chip ejector lever or button, use it if it's available). Place in new chip, being careful not to bend any of the little pins. Squirt some thermal gel on top of the CPU, bolt the fan back down onto it, and you are golden.
The easiest method to "destatic your hand" is to grab hold of a bare metal part of the chassis before you start messing with the guts of the computer. Personally, I make it a habit to touch bare metal in the case before doing anything to the hardware in it just to be safe (after 20 years of computer support, I've what static can do to a computer ... sometimes nothing ... sometimes you gotta replace everything ... mostly you just have to replace the part that got "zapped").
You are very correct. The voltage off a carpet spark you can't even see can still fry a circuit in any exposed board it hits, killing the part.
Touch either a door handle, the EXTERNAL metal of the case, or spend a couple bucks to buy an anti-static wristband. If you have to get up in the middle of working on the machine, do it again EVERY TIME you sit back down. In any case, experimentation with your computer, carefully done, can save you TONS of money over the long run. Not to mention being able to order a part and slap it in yourself gets you maybe a 3 day turnaround on a new part, as opposed to some of the slower chopshops that can take a week or more.