Yes. That is what you should do.
For your (the grantor's) 1040 enter each 1099 as you would manually enter any other 1099 of the same kind. You can list the payor as the trust name and if you want include "grantor statement" but I don't think anyone is likely to be looking at that detail.
File a blank 1041 except for filling in Boxes A-F and signature & date. Include a grantor statement. You can't use the regular TT for this, though you can use TT Business (not Home & Business) if you purchase that.
Attached is a blank example of a grantor statement. There is no special form. It can be any form that meets the 1041 instructions. See page 13-14 of https://www.irs.gov/pub/irs-pdf/i1041.pdf, which says:
Attachment. On the attachment, show:
• The name, identifying number, and address of the
person(s) to whom the income is taxable;
• The income of the trust that is taxable to the grantor or
another person under sections 671 through 678—report the
income in the same detail as it would be reported on the
grantor's return had it been received directly by the grantor;
and
• Any deductions, credits, or elections that apply to this
income. Report these deductions and credits in the same
detail as they would be reported on the grantor's return had
they been received directly by the grantor.
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