Although my Pages v14.4 works unproblematically on Tahoe 26.2, you need to understand that Pages has not (yet) been updated to work with any release of Tahoe, save for the forthcoming 26.3 and the Pages version updated for the Apple Creator Suite to be available on Jan 28. Since Pages is now nearly ten months since it was updated, this may or may not be a contributing issue depending upon where you are attempting to save those Pages documents and anything else you may have installed that may cause Pages (or System) interference.
My Mac is backed up by Time Machine to a locally mounted SSD every hour. I consider that mandatory and not just for Pages. Should Pages hang, without any prior save effort, all content will be lost, nothing can be recovered, and Time Machine would have had nothing to back up.
I only save to my local drive, or less frequently, iCloud Drive. The first thing that I do before I enter any document content is manually save the document with a proper name. That initiates autosave and versioning to protect any subsequent entered content. Apple likely has not tested Pages with any other third-party cloud service than Box, which it uses for collaboration. Saving documents to any other Cloud provider, including Microsoft OneDrive, Google Drive, Dropbox, etc. may prevent Pages autosave from protecting your document — and usually it will throw a dialog informing you it cannot autosave to that destination. Continuing to enter content is a caveat emptor at that point as it is subject to loss.
Pages is sensitive to font cache corruption, as well as anti-virus products, or so-called Mac cleaning applications that may inadvertently remove content from within the application. That makes it go berserk and hang or just crash. It is bad practice to leave Pages open with a document for days on end as some have done.
Sometimes, booting Pages into Safe Boot mode, which rebuilds the font cache and clears System caches can mysteriously restore Pages to normal behavior on a subsequent normal reboot — provided there is no other contributing factor such as other software interference or a failing storage device…
Finally, when Pages crashes or hangs, the very next launch of Pages should be accompanied by holding the shift-key. That prevents Pages from reloading its last saved execution state, and may avert continuing crashes or hang behavior.