Should I file an LLC with my domestic partner as independent contractors?

You were told wrong. The income tax will be the same amount with or without an LLC. But if you form a joint LLC with your partner, it will complicate your filing - if for some reason you want to have an LLC, each do your own. But don't create them unless you know why you're doing it. It won't save you taxes

Out of curiosity, WHO suggested you form an LLC? By any chance was it the hot air balloon company?

The reason I ask is that when there is a possibility of employee misclassification, the payer often suggests you form a business (and an LLC wouldn't be good enough--it would have to be a corporation) so that they can avoid issuing 1099-Misc forms in either the SSN of each of you or the EIN of any partnership you form.

As for income taxes, HAH.

If you form two single member LLCs, the payer is required to issue each of you a 1099-Misc for the amount each was paid (and it probably should be a W-2). Each of you would file a 1040 with schedule C and SE. Each of you would be required to pay quarterly estimated taxes.

If you form a multi-member LLC, the payer is required to issue the partnership a 1099-Misc under the partnership's EIN. You, in turn, would file a 1065 with 2 schedule k-1s. You would each "receive" a k-1 from the partnership, then file a 1040 with schedules E and SE. Each of you would be required to pay quarterly estimated taxes.

If you incorporate (say an S-corporation), the payer is off the hook for the 1099-Misc, but you become responsible for issuing yourselves W-2s. Every time you pay yourself, you would remit the payroll taxes to the IRS with forms 941. You would have to file a 1120S with schedule K-1s. You would issue yourselves the W-2s in January along with the K-1s. You would file a 1040 with a schedule E. Paying a preparer to do the 1120S and W-2s gets expensive....

In the event of an audit of the payer, the IRS may determine you were employees all along and force them to issue W-2s and you would then amend all returns to make them consistent.

What you really need to do is determine if you are really an independent contractor. If you have to use the employers equipment at times specified by the employer and follow the employer's directions then you are an employee. Earnings of $1,500 per month are hovering around minimum wage which is something a genuine independent contractor is not going to do. I suggest you contact the Arizona Department of Economic Security and ask if they think you are being treated properly then take it from there.

and LLC, a Limited Liability Corporation, is a state designation, is not recognized by IRS other than a sole proprietor, a Corporation or a Partnership

knowing as little as you do about what is involved with forming this kind of business you would be better off to merely file as a sole proprietor

Highly unlikely that it would help you as the LLC is a pass-through wrt your tax liability.

And BTW, "we aren't legally married but" means that you are both single.

It won't change anything with respect to your tax liability.