Not something I've experienced with Pages, but it sounds like something that could be caused by an accumulation of (digital) crud, for lack of a better term.
Here are two similar clean-up routines that may solve the issue:
1 A 'Shift down' launch of the application.
Save your open documents, close them, then quit Pages.
Press and hold the shift key down as you click the Pages icon on the Dock to launch Pages.
Keep the shift key down until the launch is completed.
The pressed shift key prevents Pages from reloading documents it was working on when it was closed, and runs some cleanup processes.
Open a document and test to see if spelling and grammar still have the same effect.
If the issue persists, move on to an OS restart into Safe Mode.
Like individual applications, the OS accumulates 'Crud' over time.
A "Safe Mode" restart does a more thorough set of cleaning routines. You'll notice that it is slower than a normal restart, and you will see a degradation in video performance while in safe mode.
With the Mac running, go to the Apple Menu and click Restart.
As soon as you hear the startup chime, press and hold the shift key.
When you see the Apple logo on screen, with a progress bar below it, release the shift key and wait for the startup to complete.
While in Safe Mode, test Pages for the issue you are experiencing.
After testing, do a normal restart, and test again. If all is clear, carry on. If not, I'd recommend direct call to Apple Support, or an apppointment to the genius bar if there's an Apple Store near you.
Regards,
Barry