My 7735X shaver will not hold a charge. Can I replace the battery?

You can repair this by replacing the batteries. You will find the correct batteries for model 7735X at shaveroutlet.com. See link below

http://shaveroutlet.com/shaver-replacement-parts/shaver-battery-pack-for-most-norelco-more-p-278.html

To replace, take off the plastic lotion packet cover on the back. Remove screw to take off the 'cap' on the botton then remove 4 screws to take off battery cover to access batteries. For easier replacement, it is suggested that you use a modified installation technique. This is to break the tabs off of the old batteries rather than unsoldering them from the circuit board. This allows the old batteries to be removed but leaves the tabs attached to the circuit board. You can then solder the tabs from the new batteries to the old existing tabs

Norelco 7735x battery is Norelco NiCd Battery Replacement 2 packs.