Need expertise - Looking to Form an LLC in New Mexico and Foreign LLC in Ohio - Getting mixed answers from local accountants

I am looking to do a startup for a new LLC for a services business with most of the revenue earned via my website that is still under development. I reside in Ohio and will have a home office here as a single person LLC. Since I am starting this business from home until I can start generating consistent income, I don't want my home address to be available, easily at least, to be searched and connected to my business. I would like the privacy for my family so no unexpected visitors to my home. I saw New Mexico has the ability to form a domestic LLC and services to be my registered agent there with anonymity. However, since I will have a home office and a bank account in Ohio, I know I need to register here in Ohio as a foreign LLC. My goal though is also to have my foreign LLC in Ohio also to be like my New Mexico LLC where my registered agent's details are shown on the foreign LLC paperwork and my name is not explicitly shown as searchable in Ohio so I sought help from local accountants in Ohio. I got mixed messages after meeting in person with one accountant and having a phone call with an another accountant both locally in Ohio so hoping someone here can lend their expertise. I was told by both accountants that I need to have form 533B (Registration of a Foreign Limited Liability Company) in Ohio filled out. I was told by one accountant that I can get a 2nd registered agent (first one is in New Mexico) that is physically in Ohio to sign the paperwork (without my name or address) so I'll just have to pay 2 registered agents each year (New Mexico and Ohio). The other accountant told me I have to personally sign the paperwork with my name and can't be anonymous. I want to make sure I do this right so why I need someone with the complexity of knowledge to weigh-in. As an aside, I am not looking to do anything special with taxes by forming in New Mexico, it's more for the privacy. I know my local bank and the IRS, for example, will have my information. I plan to pay taxes as expected based in Ohio. I was told by both accountants that I won't need to pay any New Mexico tax and will do it as a pass through entity only in Ohio since I will be located here and transacting here.