How can I form a partnership with another person in their business and change it to a dba of my current business?

It is entirely possible to form a partnership or other legal entity with another person and have it do business under your existing assumed name or tradename. However, you alone are the insured under your current insurance. You will need to have your insurance agent obtain an amendment to your policy to cause it to cover the operations of your related entity. Because this will involve a larger business and different risks than your policy currently covers, the carrier may require an additional premium. An attorney can assist you in forming the partnership (which would probably not be the correct legal entity) and arranging for these other matters.