Have I damaged this DC-DC converter?
I'm assuming that your module is wired like the Typical Application shown on the LM2577 datasheet, which I reproduce below:

As per datasheet, \$V_{out}\$ is determined by the following formula:
$$V_{out} = 1.23V (1 + \frac{R_1}{R_2})$$
To get the 1.24V that you're reading, \$R_1\$ must be 0 (zero). Assuming also that \$R_1\$ is the trimpot in your module, I would hazard a guess that either the trimpot is at its lowest value (\$0\Omega\$), or it failed short (burned internally closing the circuit as a wire). In any case, it seems that some component in the module has failed.
Can you measure the resistence between the trimpot contacts with a DMM and check if its really \$0\Omega\$? The measurements were these below, based OP's comment.

The trimpot seems to be stuck at 5.2 Kohm. Here are the readings:

\$5.2 k\Omega\$ between pins 1 and 2
\$5.2 k\Omega\$ between pins 1 and 3
\$0 \Omega\$ between 2-3.

Resistance does not change, even if the rotate the screw on top a lot.

Based on the above evidence, it looks like the trimpot is broken (failed short). If it was working, resistence between 2-3 would increase as you turned the screw, while the resistance between 1-2 would decrease (resistance between 1-3 would remain constant at 5.2Kohm in a working trimpot). Try and replace it and see if the module works again.
Also, based on this answer to this question of mine (Why some LM25xx DC-DC adjustable step-up converters have two negative connections (IN- &amp; OUT-) instead of only one ground?), both IN- and OUT- connections are likely to be shorted in your module already, so your shorting them could not have damaged the module.