Eternally is almost like a superlative. It is inmense. It lasts for an Eternity (which is a lot!). Everlasting is modest. It only means that will last continually without interruption. Forever is very generic. Perhaps it is even humble. Depending on how grateful you feel. You could say "eternally grateful" if you really genuinely feel inmensely grateful. You could say "everlastingly grateful" to make a compliment. You could say "forever grateful" simply to be polite.