I don't have enough experience regarding Melles Griot DPSS's tbh, however I suspect that something may not be right. Best case scenario is that it is running in idle as part of a safety feature. However it is a 22 year old laser so will have a older controller board. These do usually run at full power unless told otherwise. Looking up the model number I also do have some news on the specified output power. It's a 2.5W laser head, 5W( 5000mW) is literally just the max rating on the sticker and not the specified output of the head in this case. Also something that needs to be noted is the main part of the model number is 58-GES-305. The added R means this unit has been in for repair/evaluation. This is very costly, so in rare cases does this mean the laser actually got repaired. My gut feeling is this laser needs realignment.
I do not suggest that you try to repair this laser yourself. There are members on here that are experienced in DPSS realignment, and depending on the cost of shipping/time it might be worthwhile trying. Disclaimer however, it is unlikely that without professional alignment in a lab, that this laser head will ever operate at its intended spec. Some improvement may be achieved however.
There are things you can look out for yourself without any expertise or need to open the laser up.
Do you notice any irregularities in the laser emission? This is multimode head, but things to look at are:
Beam roundness (the dot should appear round or slightly elliptical, it shouldn't look irregular. Diverge the beam with a lens to make it easier to tell)
Beam divergence (Test the beam divergence. DPSS heads shouldn't have a divergence worse than 2mRad if properly collimated)
Transverse mode hopping (The head should be TEM00, you shouldn't see what appears to be a split beam/dot)
Power distribution in the spot. (Similar to the beam roundness and transverse mode, however you are looking for areas of higher intensity in the dot vs another. DPSS spots typically are brighter in the centre, this is normal. However it is abnormal disregarding known DPSS with TEM01/02 profiles to have lopsided power distribution to one side of the spot. Again diverge the dot for easier viewing)
Anything amiss will tell you whether there are any big issues with the laser head. If nothing like this is visible then it is either a minor misalignment causing your issues, or it is the control responsible.