Probably not. While it's true that you checked the wrong box on Form 8949, as long as the gain/loss amount in Column H is correct, then even if you were to amend just to change the check mark to Box E from Box D, such change will not effect the tax outcome. At this point, it does not seem necessary to amend your return just to make the aforementioned change. Thus, it appears reasonable to leave your original return as is.
The cost basis issue is an important issue. One way to verify the cost basis is to divide the proceeds from the sale of the shares that were sold to cover taxes--those were the shares that the company sold in 2016--by the number of shares that were sold. For example, if the proceeds from 2016 were $1,000, and the number of shares sold were 100, then $1,000 divided by 100 = $10. Thus, $10 would be the cost basis.
Another way to verify the per share price on the day the shares vested with you is to use a website such as Yahoo Finance. In Yahoo Finance, enter the ticker symbol for your company and then put your cursor in the calendar chart. You will see cross-hairs that provide a date and price. You can check the closing price of the stock in 2016 and compare that with what you see on your 1099-B.
The cost basis for the stock that was sold to pay taxes is the same as the cost basis for the shares that remained in your account. In other words, the basis is the same and it did not change. And yes, when you sold the remaining stock in 2020, you pay capital gains based on the difference between the sale price and your cost basis. Because you held the stock for more than one year, the capital gain would be a long-term capital gain.
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