you do not need a new solo 401k account because you are starting business in another state. the IRS rules and limits would not change. whether the state allows a full deduction for its income tax purposes for the contribution is a different matter but that does not affect federal taxation/deductibility
Does it mean, I only need to change LLC info with my solo 401k service provider e.g. Fidelity or Vanguard? If yes, then this makes over all process simple for contribution.