Based on the information you have provided, one explanation could be that the tape path of your machine is out of alignment. If so, then playback of tapes recorded with the same misalignment could be fine, but commercial material recorded on well-aligned machines would behave as if the tracking is off (appear noisy and shaky), like you have described. If the heads were dirty, then one would not expect a clear picture under any circumstances. Therefore, it is not likely that more head cleaning will help.
You can easily try to verify this using a know good VCR: Play one of your tapes on another VCR that you know works well on commercial tapes. If your tapes now look snowy and jumpy, you have your answer.
While you could seek out a professional repair, it is very likely that the cost to repair an old machine (often at least $75 for them to even look at the machine) will be more than the cost of a brand new VHS VCR, which are now under $40 at Wal-Mart, Best Buy, or on eBay.