I have nearly all my songs and tunes on computer, but it has taken me many years to build up my collection.
I put the tunes into the computer using a notation editor. Someone suggested Muse Score which seems to be a popular free choice. I am not familiar with it, but the one I use allows me to export the scores as a graphic which I import into a word processor and add the words underneath. I usually have the first verse, and chorus if there is one underneath the notation. With the rest of the verses underneath the dots. Works well for me but it's time consuming and if you have only limited time and a large collection of songs this may not be a practical approach. As a long term project it is worth considering, though.
Otherwise scanning in or photographing your fake books seems the best bet. If you can export to PDF, then fine, otherwise export as JPEG and import to a word processor is probably the next best bet. I would think scanning is a better bet unless you can set your camera up properly to take a clear picture. On the other hand the camera might be better if you can't open your books out flat easily.
I would suggest whatever gets the job done quickly initially, then once you are settled you can gradually tidy things up and build up a good collection in your computer.