Actually, replacing 2-prong outlets with a GFCI device and NOT installing new cable from the breaker panel CAN help with OP's problem. Doing this does NOT provide a true Ground, so there is no place for electrical noise to be sent from grounding shields in cables and devices, and they cannot help there. However, it CAN eliminate hte major cause of these "it shocks me to touch it" problems. As I said, that problem VERY often is because some part of the system is plugged into a power outlet "backwards" from other devices - that is, Hot and Neutral lines are reversed in the power feed to that part. When an electrician installs a 3-prong GFCI device in the wall box to replace an older 2-slot outlet, he / she most certainly will ensure that the polarity of the new outlet is correct. Further, in doing that job he / she should be replacing all of the other old outlets on that same circuit - the ones downstream from the GFCI that it protects - with new 3-prong outlets, and ensuring that those, also, have correct polarity. So if your system components all are plugged into the SAME circuit downstream from the new GFCI and with new outlets, the possibility of "backwards" connecting is almost gone. Even if you use different circuits for some components, IF those, too, are on circuits updated as above, they all will have correct polarity.
Now, there still is a way to get polarity wrong when plugging in a device IF that device's plug has two IDENTICAL prongs. Most newer devices have one prong wider than the other, so you can't do it wrong. SOME new devices have only two identical prongs, but that is because they use an isolation transformer as the first power input device, so wall outlet polarity is not an issue. However, IF you have a very old device not equipped this way, or IF you have a new device that HAD different prong widths and you filed down the wide one to make it fit into an old outlet, then you have defeated the protection of that design.
While replacing all the old wiring system in a house with 2-slot outlets is very much to be preferred for several reasons, it is expensive, and usually out of the question for renters. We recently had a small one-story 2-bedroom house electrical system "updated" from an old fuse panel and 2-wire system to a new breaker panel and higher amp raintg. In general the existing wire cables in the walls were NOT replaced, just re-used, so those circuits have no Ground leads. Only a few new circuits were added, and those certainly have Grounds in their cables. ALL of the older 2-wire circuits were done with a GFCI in the first outlet, and new 3-prong outlets in all the ones downstream.
Such circuits do have improved safety against electrical shocks from malfunctioning devices, as mikewinddale above says, but they do not have true Ground connections on their third round contacts. That job by pros cost about $5000, whereas complete replacement of all cables etc. would have been nearly twice as much, with significant work afterwards patching up walls that had to be opened to run new cables. For a large house even more. Makes me really glad I completely re-wired our house the new way when we bought it 50 years ago.