Electronic – Connecting negative terminal of DC power supply to negative terminal of other DC supply

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.

In general it's desirable to allow the relays to switch as quickly as possible to maximize the life. If the (protection?) relays 3 and 4 are welding on due to slow turn off, you can use a more sophisticated circuit to switch them. I suggest a power supply supervisor chip driving a transistor (see below), which will snap on an off abruptly.

Presumably relays 1 and 2 are being switched more frequently and are failing by wearing out. In that case, you should put a resistor or zener in series with the diode to allow the voltage to rise to a good portion of the rating of the drive transistors (taking care not to exceed the SOA). A zener across the transistor is another method.

Keep in mind that any circuit has some inductance and you may not be able to achieve the ratings shown on the relay datasheet even if they are switched properly. You are also expected to break the wash seal on many relays to allow the 'stuff' to vent.

A link to the relay specifications would be useful. If they are cruddy T90 style knock-offs, you may not be able to achieve the stated life even checking all the boxes off.