Adding a sheet in an unprotected workbook that has a protected sheet

Moderate Excel user but I haven't figured out what I am doing wrong here. I have Excel 2010 and a workbook with a dozen or so tabs/worksheets. On one of the worksheets, I set up a number of formulas and protected that sheet so the formula cells couldn't be changed by anyone, only the manual entry data that those cells need. I want to insert a new sheet into the workbook (actually copy one of the other unprotected sheets to a new tab as a template) and it won't let me add or copy any of the sheets. I even unprotected the one protected sheet and it still won't let me do it.

What am I missing?

Not using any code. Tried two different things; in the workbook, there is a "new sheet" icon tab at the end of the existing tabs (worksheets). I try to select it to start a new sheet and it will not let me select it. Then I have tried to right-click on any of the existing sheets/tabs so I can select Move or Copy, but that function is grayed out.

It acts like the entire workbook is protected but it isn't. Anyone can open it without a password. And except for the one sheet I do have protected (which is only to lock certain cells), anyone can go into any of the other sheets and edit them as much as they want.

Never run into this problem before.