I've recently been in 15min. timed games where my opponent will make a few moves and then seemingly disappear.  There may be twelve mins. left on the clock but there are no more moves the rest of the game.  I'm not familiar of any way to opt out of the game without it negatively affecting my rating.  I've been playing  this site for awhile and  understand this may happen occasionally.  Recently, it is has begun to occur more regularly. 

I'm a casual chess player who enjoys playing a few games. I will never beprofessional but I do like to play rated games. I'm  asking for help to find an alternative to letting the clock run out.

You could play  another timecontrol, for example 1 or 2 minutes plus 20 seconds per move.  Which should mostly get you around the 15 minutes

Otherwise there is no alternative, because simply said the line between "disappearing" and just "thinking" for a long while is to vague and offcourse differs per player and position.  

try maybe making friends who you play that don't do that,and are always online