Unless your home stereo and car stereo support CDRWs they will not play them properly and will appear to skip. Preddy much all cdrom drives support CDRWs and most DVD players do as well. Have a look in the manuals that came with your stereos and see if they mention anything about CDRW playback.
So just burn the songs onto CDR's instead which cost as little as $1.40 each. These will almost definitely play in your CD players.
Your method of burning cd's is all you need to do.