Is advance wars an obscure strategy game?
Anyway sound effects overlaid and song not finishing are two of the big perils of loop back recording. If you can find a sound test menu then do that.
After that you have three approaches
1) Somewhere in the emulator there will be a sound channel selection option.
If you are lucky the sound effects will have their own channel and disabling that channel. For the GB/GBC you would have to be very lucky indeed -- its audio is nice for what it is but compared to things like the megadrive with separate chips and GBA or DS' more general purpose approach then... yeah the best audio peeps tended to use everything to the fullest.
2) You blank the sound effects somehow. Either stop them playing, or reduce their volume to 0.
Stopping them playing tends to be an assembly hack (not usually a beginner friendly thing, though doable enough and if you are already used to BGB then you have one of the better debuggers for the GB/GBC). For say a jump command you will tend to see the sprite swapped for a jumping sprite, the character moving up in the screen and the sound played, as well as maybe some kind of hit box change. Anyway the moving on screen is fairly easy to latch onto.
Volume reduction can be done if there is a volume command, or you replace the sample (or redirect to a sample) with 0 volume, or maybe if there is a sample loop option loop the silence in a sample for as long as is necessary).
Redirection also works well if you have say a point in the game where the music is endless and want a track that otherwise gets cut short. Here if all the songs in the game use the same basic engine (even if custom to the game itself) there will usually be a command saying play track at location X, you change the numbers for X (usually with a pointer) and maybe any associated instruments (think instruments on an electric keyboard -- sound font, sound bank, instrument library... all phrases that sound rippers and editors in games use).
3) You do a straight up rip a la the audio playback set. Things like NSF, GBS, and all the other things you get to download a plugin for your audio player (usually foobar) to play. Here they essentially emulate the audio aspects of the console to play the audio itself that you located in the game and detailed. Gets easier on newer systems that have a more software defined common audio (see something like SDAT on the DS) and harder as you go further back and things get more linked to the hardware it will eventually play back on.
At this point I should however note someone has gone before you and ripped the whole lot (gameboy wars became advance wars in the west, though "wars series" in Japan and thus has something of a following). I don't think I can link it here (copyrighted works and all that) but it is out there in the first obvious place I checked for game audio rips. Has 44 tracks there.
