Starter/solenoid input lug (do this with the battery disconnected!)
Car body to negative battery cable (follow it from the battery; it should bolt on within a foot or two of the battery)
Positive and negative battery terminals (which you've probably already checked since you charged it)
Even the slightest play in any of these connections can rob you of as much as half your battery voltage under a heavy load (such as starting). In this situation, every other part of the system can work normally, because running the engine and charging require much lower current than starting the engine. It is also quite possible you just need a new battery.
(update) OH! and if the problem goes away if you jump start the car with another car or a known good battery, then forget the other stuff and just buy a new battery.
It is basically impossible to determine if a battery has a "full charge." Batteries are measured by the voltmeter. If they show 12 volts then that is about it. However, batteries show their status by the results of their capacity. Why not buy a new one? They are not that expensive and this time of year with the cold temperatures it is THE time of year when batteries die more often. The cold temperatures affects them in a negative way. Once the new battery is installed, the car should be fine.
Sounds like the battery is shot. How did you come to the determination that the battery is fully charged? Was a test under load done?
Either your motor oil is a too high viscosity or your battery is low on amperage. To test a 12 volt battery, take it to any store or dealership that sells new ones *other than Auto Zone. Ask for a free 15 second battery *load test to see what the reserve amperage capacity is. Battery amperage spins starter motors, not battery voltage.
As a matter of fact, a battery can "appear" to be fully charged @ 12.6 volts yet have little or no reserve amperage.
It's possible that the starter motor is dying but nine times out of ten it's the battery. If it jumps starts well it's the battery.
Could be many things: Bad battery cable, bad connection at the starter or the solenoid, and your battery could have a bad cell when its under load but test good otherwise.
It may still be the battery. Try jumping it or leave it on a charger overnight and try to start it in the morning with the charger still on. If this doesn't help, it very well may be the starter.
Battery just had its day, buy a new battery.
Still the battery. Gives out 12volts but few amps. It is shot. Buy a new one high CCA
Weak starter will do this. Most parts stores will test it for you.
