Electronic – I am having an issue with relays sticking

I am in agreement with @BruceAbbott's comment, above. I suspect that although your static load current is well within the relay's capability, the surge current when it closes, probably due to charging caps and/or firing up an SMPS, is welding the contacts. I suspect this because I have seen precisely the behaviour that you're describing, down to the sharp tap on the relay body causing the contacts to release.

My solution was to cause the device that I was switching (that was powered by an SMPS) to soft-start using an LCR network between the output of the relay and the input to the device.

If the cause of your problem is SMPS inrush current, and you have control over the SMPS, you may be able to put it into a "soft start" mode, either because it has the capability and just needs to be configured to use it. Alternatively, try my solution. Apologies, it was 10 years or so ago, I can't recall exactly what I used; a small series resistor (<10R) and large inductor (100uH?) and various smallish caps (1-10nF) to ground sounds vaguely familiar.

To expand on the comments, which are pretty much the answer:

The cable has a specific characteristic impedance. This impedance is created by the capacitance between wires and the inductance along them. It is in effect caused by the number of twists, the tightness of the twists, the insulator properties, etc.

Taking apart one pair for 10 to 15mm for the purpose of crimping it into a connector, which is then also designed to give the best possible conductor-to-conductor distance, is pretty much okay, no major changes there.

It is a very different thing to tear apart a wire and put an entire relay into it. You are changing the capacitances and inductances in the cable so much that the impedance changes at the point where you where you put the relay.

This change in characteristic impedance works like a mirror to part or all of the high frequency signal, where some or all of it gets reflected back to the sender. This confuses the sender (or even damages it, but modern network gear is pretty damn well protected, so it'll be unlikely), causing it to cancel the package it is sending, because it thinks it is seeing "collisions" with other data.

So every packet sent on either side comes back from the point of your relay and causes a collision detection and abort.

It is why so many HF designers are so very well paid. It's a very complicated job to make every PCB trace, every connector and every wire so that a 100MHz to 10GHz signal gets through okay. And your relay most certainly was not designed by a HF engineer.