Just to add my $0.02, I replaced the igniter first. It worked for the first few cycles, but I knew better than to get my hopes up. After 10-15 minutes, it just glowed with no ignition. I'll replace the coils tomorrow (I bought them both, since the parts store is about 45 minutes away).
One thing I noticed: when it stopped working, I would hear a rattling noise when the igniter was glowing. I wonder if it's the valves chattering. It might be a clue to help indicate whether it's the igniter or the coils (not all of us have clamp-on ammeters).