Sounds like you may be a new OMS resident. Welcome to our world.
Here's how I do it:
1) Get an xray to identify the offending tooth. This is especially important in people with crap-mouth who have many rotted teeth. You can usually identify a PA lesion on at least one. If there are several rotted teeth in a row, I will often just take them all out while I am there. Don't forget your differential diagnosis...there are lots of other things (non-odontogenic) that can cause facial swelling. Bug bites, infected furuncles/pimples, sinus disease-->periorbital cellulitis, clogged salivary ducts, etc.
2) Anesthesia. Local anesthetics don't work in the acid environment that infections create, so proximal nerve blocks are best when you can do it. For the maxilla, I like going straight up the greater palatine canal to hit V2 in the way back. You can do an extra-oral V3 but you have to know you're anatomy.For neck and mandibular angle anesthesia you can hit Erb's Point about 6cm below the external auditory canal where it wraps around the SCM. If you're fortunate to be in an ER setting where they have an IV, I sometimes supplement with 4-8mg morphine and 25mg phenergan (for morphine-nausea and added sedation). Here's a great article you can look up showing some of these nerve blocks:
How to block and tackle the face. Plast Reconstr Surg. 1998 Mar;101(3):840-51.
3) I learned in dental school to take out the tooth first because you can sometimes establish drainage through the socket. I don't like that because I still think most of these deserve a formal I&D in the soft tissue. If you have a bubble of pus you might "deflate" it by extracting first. Then it's tough to find the infected space when you're dissecting through soft tissue. Therefore I drain the pus first with an incision, then extract the tooth. But this is just personal preference.
For the incision, only incise through mucosa/skin, then put your blade away so you don't chop up the deeper structures. The rest of the dissection should be with a blunt instrument (hemostat) so you minimize risk to nerves/vessels. Push in the hemostat and spread it, then withdraw it while spread. Remember to insert it closed only. If you insert it open and then close it you risk grabbing something important. Keep aiming for the pus-pocket and have the suction ready because it usually gushes out when you find the sweet spot. Abscesses tend to form loculations so make sure you explore the entire pus pocket with you finger or instrument to break them up. Then you irrigate the heck out of it with sterile saline...one old saying is "the solution to pollution is dilution". Make sure you rinse the abscess clean with a blunt plastic-tipped syringe or something. If you put in a drain, push it to the depth of your dissection and put a stitch at the incision to hold it in place. I usually just cut a finger off of a sterile glove to use as a drain because I can never figure out where we keep real drains.
In general, it's not good to incise over the mental nerve between the 2 lower premolars. Watch out for the parotid duct. For the submandibular space, remember that the marginal mandibular branch of the facial nerve runs within 2cm of the inferior border of the mandible. A rule of thumb is 2 finger-breadths below the inferior border will put you in safe terrain. Also, when you're cutting skin, make sure to prep the site with betadine or whatever you have. I also inject local with epinephrine in the skin to minimize bleeding. Once you incise through skin and subQ fat, undermine the skin a little to expose the platysma. I like to puncture through it gently with a hemostat, undermine along my skin incision, then poke the hemostat tips back out. Now you have a strip of platysma overlying your hemostat and you can sharply incise down to your hemostat which is protection deeper structures. Once you're below the platysma, you're in the area you need to be to find your submandibular pus ball.
This is just how I do it. I would be interested how the other OMS guys around here do it differently. Maybe I can learn something.
I hope this helps.