From the described symptoms, it sounds as though a phase is down.....typically, a 12V system will charge and regulate to~14.2 V
a rectifier failure will bring it down to~10 volts either way, it's likely there won't be the voltage or current output to be useful!
You have your "old" battery? charge both it and the new one, use as needed.....a half-competent Sparks or Aircraft engineer should be able to give you guidance on current consumption for lights, radio, transponder, instruments etc. and you can then work out a sensible usage without killing the battery.
remember that flight-instrumentswith Gyros tend to use a fair bit (motor internally) Electric flaps/gear are greedy as well, but only for short periods....do things manually wherever possible.