That is entirely in the hands of your DM
The game term for a magic weapon that can talk is a Sentient Weapon. These are described beginning on page 214 of the Dungeon Master's Guide. However there is little to guide their exact nature:
Such an item might be possessed, haunted by the spirit
of a previous owner, or self-aware thanks to the magic
used to create it.
So, depending how the sword has been made (as determined by your DM) there could be a soul trapped in it, which could have a soul to be restored to its body by spells like true resurrection. However from the description of Dawnbringer (Out of the Abyss page 222) it seems it was merely forged and given consciousness and therefore has no soul to be transferred or rescued.
There might still be a way for you to do this though, but it hinges on a lot of DM fiat. The closest solutions in RAW is probably creating a golem, a homunculus, or use awaken and letting Dawnbringer be wielded by and controlling the made/awakened creature.
Creating a golem (see Manual of Golems on page 180 of the DMG) is expensive; starting at 50,000 gp and 60 days for a flesh golem and the obtained creature is probably more powerful (CR 5) than a DM would let you have.
Creating a homunculus using create homuncolous a 6-th level Wizard spell from Xanathar's Guide to Everything is a possibility cheaper option. However RAW it only creates a tiny creature which is going to have a hard time with a longsword. How to scale that up to something that can be useful to Dawnbringer is up to your DM. This is costly on the hit-points on the caster though if you want the homunculus to have more hit points.
For awaken you would need to find/make something that could wield Dawnbringer if it was sentient which might be tricky, although personally I would allow a lightsaber-wielding-sentient-bush on 'rule of cool' or maybe rather 'rule of silly'.
However your DM might be persuadable to let you build something lesser (animated armor springs to mind) to create such a vassal. However even if constructing a lesser construct for the sword it would still be perfectly reasonable for you DM to require the investment of a lot of time and money (or other exotic resources and maybe make an adventure out of obtaining those) for essentially adding another creature to your party, as increasing that number can warp combat significantly and make it harder to balance such (an thus harder to make fun). This latter point is likely why the game's core resources (i.e. the books) do not readily provide a solution for you.