It's unlikely that only one capacitor went bad. I have a creek amp, and it has a large bank of capacitors, so I'm guessing yours does, too. It's also possible for one bad capacitor to cause other things to go bad. Or for other things to cause one or more capacitors to go bad, and maybe start a chain effect.
I'd pull all of the capacitors and test them, or even do the drastic step of replacing them all. I'm not so sure I'd want one 3300 in a bank of 2200s. I'd also check the rest of the supply for problems. Other capacitors, resistors, and the diodes.
The symptoms you speak of--working for a minute then getting muffled--are consistent with the time-constant characteristics of (bad) capacitors, so it makes sense to stick to examining the power supply for now. (But see the afterthought at the end of this post.)
It's extremely unlikely that you damaged the transformer. Those things are usually self-limiting, and very sturdy. It does happen, though. They're straightforward to test. I'll give you the basic idea of a more thorough test, but it's more of a thought experiment than a procedure you want to follow. In practice, I think you're more likely to test it by just hooking up the heaviest load (lowest resistance) and doing an initial check for voltage, then checking the voltages over time while monitoring temperatures. They take a while to heat up to maximum, and the output voltage will change with temperature. The standard is for temperature to not increase more than 20C for small transformers and 30C for large transformers, and doing the test for at least two hours. Choose a load resistance calculated from the secondary output voltage and maximum current draw expected out of that secondary. It's not safe to use the speaker load and amplifier power rating because those numbers are made up more for marketing reasons than technical specs.
Obviously, test it disconnected from the power supply and hooked up to a dummy load. First, check that the output voltages are correct at very light load--they should be at least the rated voltage for the transformer, probably higher, and any pair of outputs meant for a stereo pair should track closely. Then you examine the output current as you slowly increase the current (lower the load resistance) to the maximum it can handle. The voltages on matched pairs of outputs should track or be the same across the load. Voltage output should increase steadily while (roughly) following ohm's law, and only begin to drop off as you reach the current limit. At that point, the ability to handle a magnetic field is saturated, and it can't handle any more, so you won't see much increase in voltage as you lower the load resistance. Stop there or you could damage the thing, and don't keep it there for too long. In fact, you want to do this part fairly quickly to prevent it from overheating.
The point of saturation is one of the things that can go bad in a transformer, so it's an important part of the test. You can actually change the ability of the core to handle a magnetic field. The other thing that can go bad is a short, which can be on one channel or on the input.
If there's more than two outputs (they could have from one to many), the same principle applies--stereo pairs should be the same across the load, and all of the current outputs should increase linearly, (roughly) obeying ohm's law as you lower the load resistance, until the point of saturation where they decrease. The voltages should also be what you expect from the specifications. Of course, you'll need a suitable load that can handle all that current. The (roughly) part is that the voltage output is actually dependent on load, so you get higher voltage at light loads, and less at heavier loads. A graph of current vs. load should look roughly like a straight line, but not quite the straight line Ohm's law suggests. In other words, don't look for perfection--most materials don't work that way. 10-15% variation is the norm for the vintage equipment I'm used to, and while I'd hope things were more straightforward today, there's no good technical reason they need to be better than that for a transformer.
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Speaking of temperatures...
You might want to post a picture of the supply or a schematic if you have one. It might be that some other component is going bad when heated. A thermometer (which could be as simple as your finger) and a can of cold spray would help you track it down. For starters, touch the capacitors. They should never get hot, just a wee bit warm. I don't know the temperature numbers, but I'd guess they're not even as sturdy as transformers, so anything more than 20C over room temperature would be suspicious to me. The caps in my Creek amp do run hotter than any other amp I've touched. I'm thinking heat is a more likely suspect than time constants. It's close, though--the time constant for those large caps might be on the order of a minute, so anywhere from a few seconds to 10 minutes is suspect.
