I never receive the annuity income form for my moms property until Aug or Sept. And I have not sent in an amended return get for 2018. Can I claim both of those now, on this return, on separate lines of course. I really do not want to continue filling out an amended return every year. The amount is usually $3200.00 or less per year.

Unfortunately no, you cannot report both the 2018 and 2019 amounts on your 2019 return if you received the amount in 2018 and will receive the statement that shows that.  The IRS receives the same statement you do, and will eventually notice that the 2018 amount is not on your 2018 original or amended returns, and will make the adjustment themselves. They will also do that on your 2019 return as well (since the statement you get for 2019 will show only the 2019 amount).

It is best to file an amended return to report the exact amount once you receive the statement. 

First, make sure that you are reporting the income shown on the Schedule K-1 (Form 1041) on the correct year's individual tax return of yours.  The income is reportable on the individual tax return for the year that contains the end date of the estate's income tax year.  Receiving the Schedule K-1 in August or September might mean that the estate is requesting a filing extension each year or it might mean that the estate is using a fiscal tax year rather than a calendar year.

Assuming that you are receiving the Schedule K-1 that late because the estate has requested a filing extension, it would be reasonable to file your tax return with an estimate of the amount that will be reported on the Schedule K-1, then amend later to correct that entry (still, if you've already received the money you should already know the actual amount; that money should be paid to you no later than 65 days after the close of the estate's income tax year).

You can also manage your tax withholding and estimated tax payments so that, after factoring in the income from the estate, you anticipate a small refund and just request a filing extension for your individual tax return, giving you until October 15 to file.  By doing so, there will be no real downside to you filing on extension.