I was essentially testing a circuit with no load on it.
No, you were imposing a very high load. An ammeter (amp meter) should be in series with the circuit under test. For example:
----(-)-- Power supply (+)-----(+)- Amp meter (-)---|
---(-)---- Circuit under test ----(+)----------------
When in amps mode the meter has a very low resistance, and measures the voltage drop over that low resistance (probably 0.1 ohms or less). Thus putting the meter across the power supply shorts it out.
Hence the noise it makes while trying to recover. The power supply turns itself off and on until the short goes away.
Also you are lucky you didn't blow the fuse in your meter. They are normally fused in case you accidentally do something like that.
Isn't low resistance considered a low load and not a high load?
A low resistance in series is a low load. A low resistance in parallel is a high load.
Assuming your meter presented a resistance of 0.15 ohms on the 10A range (as mine does) then using Ohms Law we can see that this would give a current of:
I = V / R
I = 9 / 0.15
I = 60 amps
So, you are trying to draw 60 amps from your 1A power supply, which then turns off.
If the multimeter is rated at 10A and the power supply can provide at most 3A, then why is there a risk of blowing the fuse in the meter?
It depends on the range on the meter. Mine has a 10A range and a milliamp range. The 10A range is fused to slightly more than 10A (naturally) and the milliamp range has a maximum of 400 mA. If you had tested with the milliamp range you would have blown the fuse. Yes, it would survive on the 10A range, if your power supply cannot output 10A.
There will be lots of web pages about measuring amps with a meter, here is one.