I want to rent a storage garage to run my business[Colorado]

I am running an eBay store to generate cash flow for my contract design company and I am in desperate need of more space, so that I can buy the bulky vintage high end audio equipment that I come across and that I am good at fixing up to resell. I am a registered LLC with the state of Colorado and I have my business checking freshly set up and ready to go. So my question is: do I start using my business name to fill out documents online and in person, or does it not work that way and I have to sign up for rentals and such with my real name and just use my companies checking account to pay the bill(s)?

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