The shield should provide two benefits:
First, I think that the shield should have its own stat: chance to deflect. Successful deflection causes the total negation of damage.
Second, it should also add a chance to absorb damage. Upon success, a fixed amount of damage is absorbed before the shield falls apart and needs repair.
The shield's use should be determined by the quality of the shield itself as well as the user. A swordsman with high dexterity should be able to use his shield to absorb the impact of enemy blows quite often, allowing him a higher chance to absorb damage with the shield. It should also increase his chance to perform a 'deflect' which completely negates an enemy attack and the shield does not take any hit to durability when this occurs.
The chance to absorb damage occurs before the chance to deflect. So, if you roll the die and you manage to absorb the damage with the shield, you should roll another die that determines if you completely deflect the attack.
It should be different from dodge or parry, though a spiked shield should probably add some chance to parry with the damage of the spikes on the shield in addition to the damage of your weapon.
The type of shield you equip should also determine a variety of things. Using a buckler wouldn't reduce your ability to dodge, and it would also have a much higher 'deflect' rate than its ability to absorb damage which isnt much, since it only covers the length of your forearm. Whereas using a tower shield would allow you to absorb a massive amount of damage, but itself would have a very low shield deflect rating, and in addition to that it would also detract from your dodge rating (as you said, it's hard to jump with a tower shield).
Using a medium shield wouldn't detract much from your dodge rating (just a bit) and it would allow a reasonable deflect rate (not as much as a buckler though) and a reasonable amount of damage absorbing ability.
I think that in addition to shields you should also add off-hand weapons like swordbreakers that add a chance to break your opponent's weapon in addition to parrying, or whips that add the ability to disarm the enemy. A jitte (japanese night-stick sort of weapon), nekode (claws), or battle gauntlet would allow you to deflect blows via parrying with your off-hand though they would not have the ability to absorb damage as a shield would.
Of course, the difference between using a shield and using a parrying weapon is that a shield would only allow you to deflect a blow or absorb its damage, but it wouldn't be able to do an attack in return unless you're using a Buckler or a spiked shield, unlike a parrying weapon which allows you to completely deflect an attack and produce an attack in return.
It'd allow for some pretty nice variations on equipment.
But as Whip said, it'd be best if critical hits bypassed any chance to deflect, absorb or parry.
If you don't have a shield you can wield two weapons (which, from what I've seen in SCA sparring, is mostly useless) or a two-handed weapon (which is useful largely due to its length).
That's only because people in the SCA are lousy at real arms combat, and they only know the basics of fighting with a single weapon (or a single weapon and a shield). Learning to use two weapons to fight requires ambidexterity (or to a lesser degree, lefthandedness) as a prequisite and years upon years of training. True swordsmen like the kensai (Sword Saint) Miyamoto Musashi were very few and far apart could handle two weapons with ease, often wielding the Katana (a two handed weapon, by default) with ease in each hand. Spanish and French stiletto fighters would often use whips or short paring knives (yeah, we use them to cut vegetables these days) on their off-hand to parry, disarm or hamstring their opponents.