I have two unconnected drain lines in the kitchen. If I start using one, can I just put a stopper on the other?
It sounds to me like you don't have two unconnected drain lines, you have ONE drain line that starts at the fridge and ends at the sink.

If you stopper the sink end its just going to fill up the drain line and back up into the filter.

You need to install a tie in for the 3/8" line under the sink before the trap.  Do not hook it in to the main drain after the trap or you'll allow sewer gases into your filter.