I am new to the forum as I needed to figure out the best way to deal with the spiderwebs in my paint just like you.
My car had swirls/spiderwebs like yours after I let an airport car detail wash and wax my car last year. It probably was due to either poor wash technique or the guy waxing the car got dirt on the applicator and used a circular motion to apply or remove the wax.
The spiderwebs are now gone by first using clay and then the Wolfgang TSR, FG and Sealant combination with a PC.
Below are a few techniques that I was taught about 10 years ago by the guys at OG and some that I picked up over the years that may help you minimize swirls after getting them removed. It may be overkill but work for me.
1) Work clean (I use a four bucket method, but I only have 1 bucket that fits a grit guard.
a) Bucket 1 for wheels and tires (do not use on your painted surface)
b) Bucket 2 with grit guard for paint
c) Bucket 3 for rinsing my wash mitt before getting more clean soapy water (I rinse my wheel brushes with the hose or dunk them in a the rinse bucket before getting clean soapy water.)
d) Bucket 4 to rinse my water sprite or MF towel when drying (I rinse out the towel in a plain water rinse bucket). I had never seen the grit guard and rinse bucket method before Mike Phillip's video. Thank you again Mike!).
2) Make sure the paint surface and wheels are cool when you wash and work in the shade if possible. If working in the sun, keep the paint wet (once wet), while washing until you dry.
3) Wash your wheels and tires first with a separate mitt for the wheels. The wheels are probably the dirtiest part of a car (at least on my BMWs) and you don't want the black tire and brake dust on your freshly cleaned quarter panels.
4) Rinse as much dirt from the vehicle before trying to wash the vehicle. I try to keep water off the paint until I get the wheels and tires clean so water doesn't dry on the paint. Once wet, I try to keep the paint wet until I'm ready to dry.
5) Wash carefully with lots of clean lubricating soapy water starting with the top, then hood, front, sides and finally back. A wash gun looks like a very cool option and I hope to try one someday.
6) Wash, dry and wax with linear back and forth motions from vehicle front to back (minimize circular motions). I just got a new sheepskin mitt which is way cool. Use the recommended amount of car wash per gallon, as surfactants tend to work best within an specific operating concentration range.
7) If you can't get something off when washing, be careful about scrubbing. You may need to use clay.
8) Flood the paint with water instead of spraying as the last rinse-step before drying to sheet the water off the paint surface.
9) Dry by laying out the drying towel to absorb the water and lift up. Or use a clean leaf blower to blow off the water!!! I used a water sprite for many years but I've been recently told to use MF instead (the water sprite seems to hold more water and wrings out easier but the Waffle weave MF towel may protect better). Rinse the water sprite or towel in a bucket before going to the next section to remove any dirt that you might pick up during the drying step. If wiping, dry in a linear fashion from front to back (if you scratch the paint, they will be harder to see in the sun and won't stand out as spiderwebs)
10) A neat trick. I have been spraying OG ShowOff detailer onto the wet painted surface while drying which provides additional lubrication and additional gloss. I have not tried other QDs during the drying step, so I'd be interested to hear what/if other people have used. I recently tried a CG synthetic detailer/gloss enhancer after drying which left a magnificent shine, but I have no idea what will happen when applied during the drying step. I'm checking the durability of the WG sealant, so I have not used any QD (or Showoff drying method) yet. Just a nice wash every weekend so far.
11) If using a QD to remove bird droppings or dust, use plenty of detailer to soften and lift, a clean and soft folded MF towel (maybe lots of clean MF towels) and wipe in a front to back motion without scrubbing. Avoid circular motions that put circular scratches in the paint.
12) When applying wax by hand, apply in a linear front to back motion.
a) Different waxes go down & bond differently so application recommendations may differ
b) A rotary buffer or DA uses a circular motion, so I'm sure that polishing and linear motion waxing application is up for debate
c) The OG guys recommended that you apply the wax linearly from front to back for the first layer. If you want to apply a second layer, apply it at least 2 days later and preferably a week later perpendicular to the first layer. OG said that the wax needed time to cure before adding another layer, otherwise you wouldn't really add layers. Maybe it's something like the 12-hour rule with the WG Sealant.
13) If you drop your applicator on the floor, you need a clean one. 5 second rules don't count if you want to work clean.
14) When removing wax, wipe using linear back and forth motions and minimize circular motions.
a) You will need to change wiping surfaces often.
b) 10 years ago, the guys at OG liked folded, 100% cotton diapers, the guys at Meguires liked 100% cotton terry and now it seems that Microfiber is the way to go.
c) Buy good quality towels, whatever type you prefer. The OG cotton diapers were awesome until my wife started using them in the house. Now it MF for me.
d) I liked Mike P's video'd method to break the wax and take little bites. I would listen to him before taking my suggestions as the gospel truth.
15) Listen to Mike Phillips and the professional detailers out there. They may have better suggestions. They have their own techniques and have tried a whole lot more products than me. For instance,the recent wax thread suggested that some waxes may make it easier to wash your vehicles down the road, rather than just the common concepts of beading water, protection and making it shine!!! I had that "aha" moment last night and maybe a high end carnuba is worth the trial!
Using the above methods, I was able to get about 2-3 months of beading between Blitz wax jobs on my garaged, daily driver BMWs, but I also used a Wash and Wash car wash plus the detailer to keep up the protection. I also clayed and used OG Omega Hand Glaze to clean and add shine before applying the wax. FYI, I recently tried the OG Blue 33 car wash which is MUCH MUCH better than the old OG wash and wax which I stopped using in 2008.
Sorry, I haven't tried any of the other AG car wash or QD brands except a couple of Megs products. The Megs Gold Class was a staple wash and I hear that the Optimum is wonderful. One of the AG-offered brands could be much better but I haven't had the opportunity or cash to try them.
I'm hoping that the new WG sealant will be even easier and more durable than the OG Blitz. It sure applied and wiped off easily, and the shine and beading is very nice. The car washed up nicely after its first bath post correction, so it could become my new favorite. Haven't tried a carnuba over it yet but someday soon.
Good Luck!