Your blade is the upper body, so it does not need a contype token. Your aura is trying to connect to a lower body, but there is no lower body for it to connect to.
Your aura tissue does not have a material token connected to it, and your energy material is not fully defined. The aura could not be connected because the raws are sensitive to upper and lower case letters, and the creature's AURA was different from the body's Aura.
This doesn't give me any errors:
[CREATURE:CURSED_SWORD]
[NAME:cursed sword:cursed swords:cursed sword]
[PREFSTRING:legendary sharpness]
[PREFSTRING:shimmering auras]
[CREATURE_TILE:'/'][COLOR:7:0:0]
[BODY:BLADE:AURA]
[DESCRIPTION:An ancient and powerful sword animated by a shimmering aura.]
[BODY_SIZE:0:0:3500]
[NO_DIZZINESS][NO_DRINK][NO_EAT]
[NO_FEVERS][NO_SLEEP]
[NO_PHYS_ATT_RUST][NO_PHYS_ATT_GAIN]
[NOBREATHE][NOEMOTION]
[NOEXERT][NOSTUCKINS][NOSTUN]
[NOT_LIVING][PARALYZEIMMUNE]
[SWIMS_INNATE][EXTRAVISION]
[TISSUE:ADAMANTINE]
[TISSUE_NAME:adamantine:adamantine]
[CONNECTS]
[TISSUE_MATERIAL:INORGANIC:ADAMANTINE]
[TISSUE_LAYER:BY_CATEGORY:BLADE:ADAMANTINE]***
[USE_MATERIAL_TEMPLATE:ENERGY:ENERGY_TEMPLATE]***
[TISSUE:AURA]
[TISSUE_NAME:aura:auras]
[CONNECTS]
[FUNCTIONAL]
[TISSUE_MATERIAL:LOCAL_CREATURE_MAT:ENERGY]***
[TISSUE_LAYER:BY_CATEGORY:AURA:AURA]
[NATURAL_SKILL:SWORD:8]
[NATURAL_SKILL:DODGING:8]
[PETVALUE:100]
[ATTACK:SLASH:BODYPART:BY_CATEGORY:BLADE]
[ATTACK_SKILL:SWORD]
[ATTACK_VERB:slash:slashes]
[ATTACK_CONTACT_PERC:60000]
[ATTACK_PENETRATION_PERC:6000]
[ATTACK_VELOCITY_MODIFIER:1250]
[ATTACK_FLAG_EDGE]
[ATTACK_PRIORITY:MAIN]
[ATTACK:STAB:BODYPART:BY_CATEGORY:BLADE]
[ATTACK_SKILL:SWORD]
[ATTACK_VERB:stab:stabs]
[ATTACK_CONTACT_PERC:50]
[ATTACK_PENETRATION_PERC:3000]
[ATTACK_FLAG_EDGE]
[ATTACK_PRIORITY:MAIN]
[ATTACK:SLAP:BODYPART:BY_CATEGORY:BLADE]
[ATTACK_SKILL:SWORD]
[ATTACK_VERB:slap:slaps]
[ATTACK_CONTACT_PERC:60000]
[ATTACK_PENETRATION_PERC:6000]
[ATTACK_VELOCITY_MODIFIER:1250]
[ATTACK_PRIORITY:SECOND]
[OBJECT:BODY]
=====Aura=====
[BODY:AURA]
[BP:AU:aura:auras][EMBEDDED][FLIER][THOUGHT][CONTYPE:LOWERBODY][CATEGORY:AURA]
[DEFAULT_RELSIZE:50]
=====Blade=====
[BODY:BLADE]
[BP:BL:blade:blades][UPPERBODY][LOWERBODY][CATEGORY:BLADE]
[DEFAULT_RELSIZE:1000]
Edit: I've left stars next to the changes I've made in the creature raws. In the body raws, I've removed the contype from the blade, made it a lower body so that the aura could attach to it, and fixed the aura's ID.
To connect certain tissues to body parts you use the tag [TISSUE_LAYER:BY_X: X :TISSUE]
BY_X can be BY_CATEGORY, which will make the tissue cover every body part with the correct category type, BY_TYPE, which attached itself to tokens such as STANCE, GRASP, UPPERBODY/LOWERBODY, etc., and BY_TOKEN, which connects the tissue to a body part by the body part's ID. The final argument is the ID of the tissue that you want to be used.
Below are all examples of how to connect the adamantine tissue to the blade.
[TISSUE_LAYER:BY_CATEGORY:BLADE:ADAMANTINE]
[TISSUE_LAYER:BY_TYPE:UPPERBODY:ADAMANTINE]
[TISSUE_LAYER:BY_TOKEN:BL:ADAMANTINE]