Do I need to change car-battery or retry jumping?

You can do several things prior to assuming the battery is dead, but more than likely it is.

Put the battery on a 2Amp charge overnight. Then turn the headlights on for about 10 seconds. Then test the batter with a digital multimeter (DMM). If the charge is above ~12.5vdc, it should be able to survive, at least for a while. If it is below the 12.5vdc mark, take the battery to a local auto parts store. They can put it under load and give you the approximate life of the battery or it's actual capacity. They can tell you for sure.

The problem here is, you've let the battery sit too long. A battery will lose energy over time as it sits, but then throw a minor load from the car onto it and it will completely drain the battery. A battery likes to be charged and lives a long, healthy life if you can keep it that way. In the future, if you plan on letting your car sit, put a battery tender type charger on your car. They are relatively cheap and will keep the battery up to shape without overcharging. A "trickle charger", or even the 2A charger (most low settings on regular chargers) will overcharge the battery given enough time and will cause it to (in most cases) boil over. The battery tender won't do this, but rather will help the battery stay where it likes to be ... charged.

should I try to jump it again and drive it around for longer with all AC and radio off?

Please don't do that again (that is, jump start a car with a dead battery). If you suspect a battery is dead and not merely flat, it may not act as a good enough buffer for the car's electrical system. You should not jump start a car that may have a dead battery.

This means that when you jump start the car, and disconnect the donor vehicle, there can be voltage spikes from the alternator because the battery is not buffering those away. All sorts of expensive electronic damage can happen. In fact, jump starting a car with a dead battery is equivalent to removing a battery on the fly when the car is running. Yes, you can do it. No, you should not do it.

The correct way to start a car with a battery that may be dead is to charge it, either rapidly via thick jumper cables from another vehicle, or less rapidly via a charger.

Then, immediately when the charging is done (10 mins via thick jumper cables should be enough, with a dedicated charger few hours usually gives just enough juice unless it's a trickle charger), try to start the car. If it doesn't start, then replace the battery with a known good new unit.

Jump starting should be reserved for cases where you know that the battery is good but flat, for example when you forgot the headlights on and noticed it on the same or next day but too late.

Lead-acid batteries don't like extended deep discharge, so forgetting headlights on and noticing it after a few weeks, or letting the car sit for months, could destroy a lead-acid battery.

The battery is one of the most important components in a vehicle that requires uttermost care during frosty weather. During this time, the fluid (particularly distilled water) inside the battery gets thickened which prevents the smooth flow of the current. This leads to permanent & irreplaceable damage to the cells.

The best option to avoid this situation is to perform a Volt test to check the condition of the battery and change it, in case it is detected as weak. Indulge in battery care on a regular basis.

Car batteries go through a challenging phase during the cold weather as they now have to provide more current than usual. So, keeping a few jumper cables with you can reduce the risk of your battery running down slowly.