I haven't tried the class, but I can give some feedback. My overall points:
1) Many features look really interesting. I like a lot of it.
2) I think the class intrudes too much on existing classes, namely the Wizard and the Eldritch Knight.
3) I think the class has too many 'out of flavour abilities' and could use a little more flavour.
4) I think the class is too powerful, overall.
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I'm not too organized, so I'll just give my thoughts point by point:
Main Class
Saving throws - I think Con + Int makes more sense.
Skills - It needs some more skills. Maybe Acrobatics and Perception.
Runebound Weapon - Very similar to a level 3 Eldritch Knight class feature. Not really necessary either. Maybe reflavour it a bit, so it has more uses like light etc.
Sword and Sorcery - Seems more like a feat than a fighting style, to be honest. I would rather something like this is a class feature, than a fighting style. But I like the idea.
Spell recall - Very close to a Wizard ability, and not really that 'in flavour' for the class. I would substitute this with something else.
Dispelling Strike - I kinda like this one. Not sure if it is too strong, but I like the flavor.
Spell Reflection - I think you have an error here, where it is meant to say 'or higher' in the third line. It is a very strong ability if it works as I read it. Too strong. I think it should be a check on a spell level equal or higher, and automatic failure if you use a lower slot.
Greater Spell Access - Decent I guess.
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Swordmage
Spellstrike - Very, very strong. You basically get to deal double damage each round - 2 levels before everyone gets their 2nd attack. This need to come at a higher level I think.
Spell Combat - The Eldritch Knight's level 18 ability at level 6. Way too strong.
Hindering Strikes - As the Eldritch Knight for the same level.
Spellstrike Overload - I like this one. Balanced because of the limited use, and useful, without being overpowering.
Spell Combat Mastery - So, automatic quickened spells + full attack action? And you can even use Spellstrike with the spell you cast? This is way too strong, even for a level 20 ability. Way, way too powerful.
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Arcane Archer
Imbue Arrow - I like this. Needs a limitation to Int per short rest or can't be used with cantrips, to prevent Thunderclap spam. Or remove the 'even if it normally has a range of self only'.
Seeker Arrow - Probably needs some fine tuning, but no real issues here.
Phase Arrow - I'm not really feeling it. It's not that it is too strong, it just doesn't seem necessary or flavorful.
Hail of Arrows - Steps on rangers toes. Doesn't really have anything to do with spellcasting either. Not really feeling it, although I don't feel it is too strong.
Arrow of Death - The damage dealt might need some adjusting, but the ability seems fine enough to me.
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Overall thoughts
I think you need to rebalance the main class, and then work out from there with the archetypes. You also need to decide if this is a substitute to the Eldritch Knight (and borrow a lot more from that class) or if this is a wholly different class (in which case several abilities needs to be reworked).
The Swordmage archetype needs most work, in my opinion. It seems straight up broken to me.
Good work though, I might borrow some of it
1) Many features look really interesting. I like a lot of it.
2) I think the class intrudes too much on existing classes, namely the Wizard and the Eldritch Knight.
3) I think the class has too many 'out of flavour abilities' and could use a little more flavour.
4) I think the class is too powerful, overall.
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I'm not too organized, so I'll just give my thoughts point by point:
Main Class
Saving throws - I think Con + Int makes more sense.
Skills - It needs some more skills. Maybe Acrobatics and Perception.
Runebound Weapon - Very similar to a level 3 Eldritch Knight class feature. Not really necessary either. Maybe reflavour it a bit, so it has more uses like light etc.
Sword and Sorcery - Seems more like a feat than a fighting style, to be honest. I would rather something like this is a class feature, than a fighting style. But I like the idea.
Spell recall - Very close to a Wizard ability, and not really that 'in flavour' for the class. I would substitute this with something else.
Dispelling Strike - I kinda like this one. Not sure if it is too strong, but I like the flavor.
Spell Reflection - I think you have an error here, where it is meant to say 'or higher' in the third line. It is a very strong ability if it works as I read it. Too strong. I think it should be a check on a spell level equal or higher, and automatic failure if you use a lower slot.
Greater Spell Access - Decent I guess.
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Swordmage
Spellstrike - Very, very strong. You basically get to deal double damage each round - 2 levels before everyone gets their 2nd attack. This need to come at a higher level I think.
Spell Combat - The Eldritch Knight's level 18 ability at level 6. Way too strong.
Hindering Strikes - As the Eldritch Knight for the same level.
Spellstrike Overload - I like this one. Balanced because of the limited use, and useful, without being overpowering.
Spell Combat Mastery - So, automatic quickened spells + full attack action? And you can even use Spellstrike with the spell you cast? This is way too strong, even for a level 20 ability. Way, way too powerful.
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Arcane Archer
Imbue Arrow - I like this. Needs a limitation to Int per short rest or can't be used with cantrips, to prevent Thunderclap spam. Or remove the 'even if it normally has a range of self only'.
Seeker Arrow - Probably needs some fine tuning, but no real issues here.
Phase Arrow - I'm not really feeling it. It's not that it is too strong, it just doesn't seem necessary or flavorful.
Hail of Arrows - Steps on rangers toes. Doesn't really have anything to do with spellcasting either. Not really feeling it, although I don't feel it is too strong.
Arrow of Death - The damage dealt might need some adjusting, but the ability seems fine enough to me.
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Overall thoughts
I think you need to rebalance the main class, and then work out from there with the archetypes. You also need to decide if this is a substitute to the Eldritch Knight (and borrow a lot more from that class) or if this is a wholly different class (in which case several abilities needs to be reworked).
The Swordmage archetype needs most work, in my opinion. It seems straight up broken to me.
Good work though, I might borrow some of it
