Nintendo has two forms of revision numbers that they use for their cartridge games. The first one is the "-n" identifier on the end of the serial number which means that something physical about the cartridge has changed. In 99% of the cases this is an update to the label.
Back in the mid-90's we saw a lot of "-1" games appear because common games that everyone wanted to buy were being distributed when the ESRB was being instituted. There are many games that have a standard version and a "-1" revision simply because the ESRB rating was slapped on the label.
Nintendo's own Player's Choice logo is another good example of many label updates that required a new revision number because of the addition of the Player's Choice badge.
What you are really interested in is the ROM version of the game that is on the memory chip on the PCB. For quality control Nintendo does mark their games with a ROM ID on the outside of the game. For any cartridge game with a label on the back (NES, SNES or N64) there is an impressed number on the sticker. For games with only a sticker on the front (GB, GBC, GBA) the number is impressed on the game label, with the exception of the DS and 3DS, which also have the revision stamped on the back.
This number represents the facility/manufacturing line that the cartridge was assembled on. If your cartridge has the first version of the original ROM you will only see the 2-digit facility number. However, if a new version of the ROM was released from the developer to manufacturing, then a letter was appended to the end of the facility code.
The first revision will be an "A", the second revision will be a "B", and so on. If you take a look at these identifiers you can determine which version of the ROM is on your cartridges. If both games have no revision code, or if they have the same revision code, choosing which game to keep is up to you.
The ROM version can be a bigger deal than the serial number variation, depending on the nature of the ROM change. Nintendo has been known to adjust the art in some of their games for no apparent reason. Choosing which revision is a matter of preference.
You may be wondering where to find out information about specific cartridge ROM variations. The best place to go for a detailed log of variations to game ROMs is The Cutting Room Floor. (Beware, this is a blackhole for retro-gamer nerds, such as myself!)
To be clear, this site DOES NOT distribute game ROM files but they have thoroughly documented differences between the software variations of many games, old and new, for Nintendo and many other systems.
The only unfortunate aspect of TCRF is that they do not claim which revision ID stamped on the game goes with which revision detailed on their site. Instead, they just say "Japanese Version 1" or "US Version 2" when outlining differences of ROM versions. You may have to read the details on specific revisions to determine which ones may exist on your cartridges but that's never a difficult process.
Just so this question will have an answer, for future reference:
The problem was the A/C adapter I was using. It was the only component I didn't have two of, so the real answer is, when you're trying to eliminate the source of the problem, never skip any parts that could be the source of the problem :)
Not sure what was wrong with the adapter but it got really hot just sitting plugged in, with the SNES off, overnight. When I dug up another adapter with the same in and out voltages and swapped that last part, I got a clean signal (on every TV, every connection, etc.)
Best Answer
Natively, yes you have options depending on your model SNES and TV. Your best bet is S-Video, as that is the most common plug available, but optimal is RGB but that isn't typically supported on modern TVs.
The option you are using actually provides the worst image possible, but all of these will look bad on a HDTV, due to the way it upscales. Your better options are an external upscaler (Which is expensive), a CRT (Super clunky), or emulation.
The most effective option I have found without a having to spend around $600 or lugging a CRT into the lounge room is the Retron 5. It's light weight, uses original cartridges and controllers and upscales to HDMI. However, it is an emulation machine, and it isn't quite the same as playing on original hardware.
There is this option as well: SNES Component cable. I haven't actually seen these in action however, so research is advised.
A great site for information about getting retro consoles working in a modern environment is Retro RGB. Most of this information is from that, with a bit of personal preference from Youtube videos.