"If I added both forms, my gross distribution is double."
That's correct, both are required to be reported. What matters to you is the taxable amount that is included on Form 1040 line 5b, not the gross amount amount required to appear on line 5a. The amount included on line 5b would be only the amount from box 2a of the code-G Form 1099-R since the amount in box 2a of the code-H Form 1099-R is zero.
