It is not possible to construct a charger that simultaneously provides a fixed constant voltage and a fixed constant current. The two are intimately related; at a given state of charge (and temperature and so forth), there is a single voltage vs current curve that applies to the battery. It is closely related to Ohm's law, although the relation between voltage and current is not precisely linear in this case.
Your charger might put out 13.something volts under a small load (low current required) which tapers down to 12.7 at 15A. There are plenty of other things that could be going on, too, such as seeing the effects of an old split output converter.
At any rate, I think you would be well served to replace the charger with a modern multi-stage unit, such as a Progressive Dynamics PD9245. It is more than $50, but worth the money and kinder to your batteries. (I guess if you rely on solar nearly exclusively it makes little matter what converter or charger you have.)