In my area, there are LEAST 3 other contractors with a similar name. This is because we are all Sole Propietors. All perfectly legal.
The only way I know of to get a clear cut distinction is to change your name...from DBA or T/A Adam's Atoms, to a more "corporate" name - Adam's Atoms LLC; Adam's Atoms Inc.; etc. Of course this may change a whole myraid of items and issues for you.
I would NOT change the co. name to Adam Electric, or just use initials...I kind of like that name Adam's Atoms....too bad my name isn't Adam...LOL
. It's a clever name and has a hook to it....(I hope mine does to ~ Celtic Electric). Working up an easily recognizable name is not always easy, building a good rep. on that name is also not always easy - changing gears will lose the race, as you, basically, start all over again...no name recognition.
Talk to a CPA to see what the (tax)implications of changing your name to LLC, INC., etc will involve.
If it makes sense(cents) for you to add the initials at the end, then so be it.