Consider something like this kit from Apogee. It's linear, has adjustable current limit (i.e. important) and meets your budget, as long as you can live with 3A for the time being. If you go bigger, you can get 5A but you're looking at closer to $300.
You can also keep your eyes open on online auction sites for used industrial equipment like the HP6038A (60V 10A) or the 6Instek GPC-3060D (dual 30V 6A, can be paralleled).
Just because the continuity test function of your meter does not detect a short does not mean there is an unintended path for excessive current between the supply rails. There are many potential faults that could exist but won't be detected with a meter like that, but here's one: a forward-biased diode between the rails.
Most meters, except the very cheapest of them, test continuity with less than 0.6V precisely so that they can't forward-bias a silicon P-N junction. Thus, they will not detect a diode as "continuity" regardless of which way you place the diode with respect to the meter. However, to any power supply significantly more than 0.6V, the diode might as well be a dead short, if it's forward-biased.
If your circuit is simple, the easiest way to troubleshoot is usually to inspect it very carefully, looking for errors. If that fails, you can connect the power supply and leave it in CC mode, with the current set low enough that you aren't burning anything. Then, with a good, precise voltmeter, start measuring the voltage drop between points on your board. Where there is a voltage drop, there is high current (remember, \$E=IR\$), and thus you know the fault is somewhere between those two points. Be methodical about it and you should be able to find the fault by process of elimination.
Best Answer
How do you measure this continuity ? In the diode/beep test of your multimeter ? Now switch off the supply (very important !) wait a couple of minutes for verything to discharge and now measure again but use the Ohms range. Is it 0 ohms ? I expect that it is not !
On my Fluke multimeter the continuity test already beeps when I connect 100 ohms. So a continuity beep does not mean that it actually is a 0 Ohms connection !
Also there will be a lot of decoupling caps on that supply. When they're discharged it looks like a low ohmic connection (to the multimeter) so it beeps. After some time (could take minutes) the caps are charged and the beeping stops. Try that with a 100 uF capacitor (or larger) if you that lying around.
So no, it is not bad, just expected behavior ! In fact, this is how I would check that a supply is OK when a device is off, if I would measure 0 Ohms I would check if it might be shorted !