Assuming a player has War Caster to cover Somatic components and is holding both a weapon and shield, can they still use spells with a Touch range, such as Shocking Grasp, or do Touch spells imply a free hand is required?
For Shocking Grasp, yes
Shocking Grasp description says explicitly you use your hand to touch the target:
Lightning springs from your hand to deliver a shock to a creature you try to touch.
For other spells, it depends
For instance, the Cure Wounds spell simply states that
A creature you touch regains a number of hit points equal to 1d8 + your spellcasting ability modifier.
The general spell range rule doesn't clarify "touch" (PHB page 202):
Some spells can target only a creature (including you) that you touch.
Therefore, the outcome is up to the DM - if they tend to read the rules literally, they probably said you can touch the target with your chest or elbow. If they stick to the "healing hand" trope, they can require you to touch the target with your hand. Spell descriptions are loose by design in 5e, allowing various DMing styles. Many DMs allow inventive spell use, others insist that spells do only what they say they do, regardless of conditions (for instance, Produce Flame doesn't become "Produce Steam" undewater).
Presumably, "touch" means touch with your hand
The Paladin's Lay On Hands feature mention no hands in its description:
As an action, you can touch a creature and draw power from the pool to restore a number of hit points to that creature, up to the maximum amount remaining in your pool
It says only that a Paladin "touches a creature", but is assumes they does it with their hands, hence the name. That allows us to conclude that "touch" means touch with your hand, unless stated otherwise.
But nothing in the rules says you must have a free hand
In 5e there is no special rule like "when you cast a spell with a touch range, you must have a free hand to deliver the spell". That means, even if your hands are occupied, as a player, you definitely can announce "I cast spell X", even when this spell has the "touch" range. As Dale M said, you can touch things with the back of your hand as well. But your DM might say "no, you can't perform this spell, because both your hands are occupied". In this case, the DM would be right, because only they knows the details how does magic work in the game world.
Spells do what they say they do: if it says you must touch them then any part of your body will do, if it says you must touch them with your hand then you must use your hand.
However, you can still touch someone with the back of your hand even if you are grasping something.