Well the thing that you have to be most careful of just taking it apart and doing the job is that ESD (electro-static discharge). In shops and manufacturing facilities they make you wear a little wristband that's grounded so it isn't an issue, but if you're inexperienced and don't have a proper
workbench then it becomes an issue you need to be aware of.
As long as you're not taking it apart on the carpet of a living room or something it's not a huge scare, but yeah potentially you could accidentally touch something and fry a chip or another component without even noticing. Not trying to scare you or anything but it can happen. Doesn't even take enough that you would feel a little shock like when you get a little static shock touching metal and you can feel it.
But really dude, you already took the thing apart and identified the problem. So I would still encourage you to do the job yourself.
A quick google and I found this...
Laptop Headphone Jack Repair - What You Need To Know, which doesn't tell you a whole lot but it's a starting point. If you can see that the only issue is the one headphone jack then you should be good. If anything else around it looks f*cked up then you might be in more serious trouble and have to replace the whole board.
You already took it apart, so you know whether the jack is on the main PCB, or a sub-PCB. Then you just need to look up the brand of your computer for a replacement part. Personally, I've never had to crack open a laptop so I can't help you too much further, but I found this...
Soundboard for a Dell. You should be able to find a replacement part pretty cheap, at least cheaper than paying somebody $100+ for a 20 minute job.
Without being able to actually look at it I can't say for sure whether it's worth it to just get it repaired or if you could really do it yourself. But if it was mine I would give it a shot. Buy a soldering iron and a little spool of wire, cut some small strips of wire and peel the insulation off the ends and just practice soldering until you get the hang of it. I'm sure there's some tutorials on youtube, and it's a valuable skill to have if you're into electronics. Guitar/bass amps, stereo receivers, and other various music equipment that you're regularly plugging and unplugging from can usually be repaired for little to nothing when they get worn out.
I had an old guitar practice amp that the input jack got f*cked from people tripping over the cord when it was plugged in. Took half an hour to fix an otherwise worthless piece of equipment at that point.
I could try to help you more but it's kind of hard to repair a piece of equipment over the internet.
If you're really sketchy about it, just go for the USB adapter as you could potentially do more harm than good. You'll probably buy a new laptop in the next couple years anyways the way computers go. Let me know if I can help you anymore and good luck, learning new skills is awesome.
Peace!