Assuming the corpse is a creature, yes
The Shambling Mound's stat block is very clear that it can have only one creature Engulfed by it at a time:
Engulf. The shambling mound engulfs a Medium or smaller creature grappled by it...If the mound moves, the engulfed target moves with it. The mound can have only one creature engulfed at a time.
So, assuming you can treat the dead creature as a creature, the Mound would indeed have to drop it in order to Engulf a second creature. This makes sense within the narrative, as it is size Large and there is room for only one size Medium creature inside it at a time.
But what if it is (only) an object?
Unfortunately, the rules are not clear on whether a corpse is a creature, an object, or both. Thus it is possible that a dead creature is actually only an object, and if so, it would not 'trigger' the limitation that only one creature can be Engulfed at a time.
Even more unfortunately, it is difficult to say, RAW, what would happen to a corpse-as-object that had previously been an Engulfed creature. The prerequisite for being Engulfed is that the target has been grappled by the Mound, and the Grappling Rules apply only to creatures:
When you want to grab a creature or wrestle with it, you can use the Attack action to make a special melee attack, a grapple.
Thus, when the creature died and became an object, it was no longer a legitimate target for the Grappled Condition, and thus for the Engulfed state.
The 5e rules, in general, have a difficult time dealing with "lingering effects" - what happens to a creature, or object, when it is under an ongoing effect for which it is no longer a valid target? See, for example, this tweet by Jeremy Crawford, where he says 'there is no rule', or my answer to "Do lingering effects vanish when you use Wild Shape?", in which I try to give a summary of what we know about different kinds of effects.
Thus, if the corpse is an object, it is unclear whether it being an invalid target for Grapple would either permit the Mound to Engulf a new creature while still retaining the corpse-object, or would force it to drop the corpse-object regardless of whether or not it Engulfed a new target.
What I would do
In my own games, the rules exist to create shared baseline expectations for the players, and thus I do give great weight to RAW. But when the rules are unclear, as in this case, I decide in favor of the narrative. That is, what makes sense, and what ruling would support verisimilitude?
It seems like the narrative of a Shambling Mound is that it has space for only one Medium-sized 'thing' inside it at a time, regardless of whether that thing is technically a creature or an object. If the intent of the Shambling Mound was to kill as many of the PCs as possible, it would simply drop the corpse to make room for its next prey. If for some reason it was important to retain the body (was dragging it off to digest later, was commanded to collect the PC's by the mage controlling it, etc.), it would move the body outside to the surface of the Mound while still grasping it, and switch to the 'carrying an object' rules while it Engulfed the still-living PC. Doing so would both be a striking visual and provide access to the corpse for a party that hopefully has the revivify spell.