In Ken Follett's World Without End, depicted the mid-14th century, he writes "priests were not supposed to shed blood, a rule they acknowledged cursorily by using blunt weapons on the battlefield". Did priests actually do this in medieval times - fight wars but using blunting weapons to avoid blood?

So Follett is not precisely making this up, but it's also not historically accurate. He's thinking of Edward Freeman, The History of the Norman Conquest of England, and the many accounts that Freeman influenced:

Close at [William's] side, and armed with the same fearful weapon [a mace],
rode one whose name was soon to be joined with his own in the mouths of Englishmen, and who was to win a far deeper share of English hatred than the mighty Conqueror himself.

Odo; the warrior-Prelate of Bayeux, rode in armour by the side of his brother and sovereign, as eager and ready as William himself to plunge wherever in the fight danger should press most nearly.

To shed blood by sword or spear was a sin against the Church's canons, but to crush head-piece and head with the war-club was, in Odo's eyes, no breach of the duties of a minister of peace.

Freeman is himself citing Wace's Roman de Rou, written in 1106:

He had put a hauberk on, over a white aube; wide in the body, with the sleeve tight; and sat on a white horse, so that all might recognise him. In his hand he held a mace, and wherever he saw most need, he led up and stationed the knights, and often urged them on to assault and strike the enemy. (trans. Edgar Taylor)

Not cited by Freeman, there's also the Bayeux Tapestry, which has a picture of Odo with the caption Hic Odo eps baculum tenens confortat pueros--Here Odo, bishop, holding a stick, confronts the "boys".* William of Poiters, another near contemporary, has "he never raised arms and never wanted them to be raised."

Setting aside the question of whether or not Odo was actually carrying a weapon-weapon: as you've noticed, none of the medieval sources say the least bit about whether he used it, and more to the point, none of them say anything about a Blunt Weapon Exception. The BWE is a romanticized 19th century myth.

As to the related questions, (1) Were priests forbidden to carry weapons (2) Did they carry them/did they carry them anyway. The first answer is "It's complicated"; the second is "Yes, some of them, of course."

As Lawrence Duggan has pointed out, the only^1 real blanket ban on clerics carrying weapons is in the Decretals of Gregory IX (1234), that outright says clergy who carry arms will be excommunicated--Clerici arma portantes. The problems are manifold, namely: (1) What are the bounds of "clerici" (2) What does "portantes arma" mean (what is carrying, what count as arms) (3) The papal issuance of the Decretals doesn't really reflect what the average priest on the ground was learning. Much more important for what they would believe was licit and illicit came from local synod rulings. (Gregory takes his own assertion from a council held at Pictou.) And those are much more ambiguous, contradicting each other, and allowing for lots and lots of exceptions to Gregory's flat assertion.

As to clergy (of all sorts) carrying and using weapons? Yes, there are absolutely stories, right on down to priests using knives in barfights. (Well, tavern brawls.)

Follett is fascinating to me as a historical fiction author. When there's something he doesn't know about the Middle Ages, he does scrupulous research. When it's something that "everyone knows," he just goes by it--and it's often quite wrong. (Pillars is my favorite fiction book of all time, and I really enjoy World Without End, too...but everything about Carys's plotline is an accuracy disaster.)

^1 ETA: Okay, I kept looking and found a set of constitutions from early 1200s Paris that forbids priests from carrying "knives with sharp tips." My read on this has to do with everyday violence rather than battle: the knife can still have a sharp edge so as to cut bread and such, but not a sharp tip for impulsive fighting.