Hello and welcome. The tank the goldfish is in is way too small for a goldfish. Depending on what type, goldfish should live a very long life,grow to 10+", and need at least a 29 gallon to be even somewhat comfortable. They are also very social, so he should have one friend in that tank.
I do not understand what you mean by "dealing with ammonia spikes". Is your new tank cycled, and what do you have in it?