Usually if an electronic sniffer says coil leak, no further testing is needed.
Yes, home warranty companies are awful! Usually best to get a cash option and find your own contractor. Around here anyway, those willing to take the peanuts they are paid doesn't get you good contractors. There are exceptions obviously.
Unless you want to spend the money on a new furnace to get more efficient or variable speed, keep the 13 year old machine if it isn't causing you any trouble. As long as it blows the right amount of air, it's fine. Just remember you won't get past about 14 SEER with it so if someone talks you into high SEER or 2 stage, don't bother.
Amana has premium units (models start with A) and it has Amana Distinctions which is a basic Goodman rebadged. Clarify which is being quoted you.