Base damage is tied to Level, not to strength.
Strength simple is a Multiplier that is 1.0 at the global Level base value.
Dragoon STR should put that to somewhere around 1.35-1.4
Base damage at 50 could've been ~20
Base damage at 60 might be ~22
Then there's also the that if we assume that STR is a Multiplier starting at base - 277 is a result of your 247 "base" at 50 + 30 spent stat Points, whereas 277 at 60 is a result of your "base" at 60 + ~20-25 Points. You might also want to compare that to either +30 or + 35 if we assume that STR scaling changes.