Reddit mentions of AHA/BHA Acne Clearing Gel, Maximum-Strength Salicylic Acid Acne Treatment, Clears Up and Helps Prevent Acne, Helps Minimize the Look of Acne Marks and Pores

I can totally empathize, as it suddenly got very bad for me too. My picking issue concerns chronic ingrown hairs (some legitimate, some self-induced by creating an "environment" really conducive to their development via picking). Exfoliation is touted as the best treatment, but, well, I kinda don't think it really does dick. I have used these and also usually just over my jawline and neck. I feel like they help a little in preventing/reducing the occurrence, maybe better than nothing. I like 'em because they contain both AHA and BHA, which are supposed to work pretty well synergistically. But pretty expensive.

AHA (glycolic acid, lactic acid, mandelic acid, etc. in a variety of concentrations and formulations) is good for surface stuff like whiteheads and uneven skin tone. It will burn like mad for a few seconds if you use it over any open wounds (and possibly even if you don't). Also pretty drying. You'll be red and flushed, so moisturizer and SPF is a must after applying anything with an AHA.

BHA (aka salicylic acid, like Stridex pads) penetrates deeper into the pore. Good for blackheads and reduces inflammation. Not nearly as intense as AHAs (you rarely find concentrations of more than 2%) but still a bit drying in my experience. My skin tends towards dry anyway, so I always have to use a moisturizer.

There's some controversy over whether physical or purely chemical exfoliation is better. I've heard of the medicated pads having the potential to make "micro tears" in your skin, leading to breakouts but that really smacks of some bullshit. If you find that physical exfoliants are too harsh, don't use 'em. Whenever my skin seems to react badly to the pads, I tend to use this gel. Paula's Choice does have some nice chemical options in the form of leave-on gels, liquids, and creams, but like you said, expensive as fuck. Honestly, drugstore/generic equivalents will probably get the job done without breaking the bank. The fewer ingredients, the better (as a rule of thumb).

I've learned that keeping my wounds from drying out and scabbing over is key to speeding healing and minimizing scarring. Occlusives like Neosporin, Aquaphor, or even plain Vaseline are your best bets. Apparently you're not supposed to use Neosporin for more than a few days, but I used to slather myself with that stuff for weeks at a time before discovering Aquaphor. Didn't have any issues (except perhaps more resistant bacteria? Who knows). People also swear by hydrocolloidal bandages (think blister patches). They promote a moist healing environment w/o having to use any chemicals and you can leave on for several days.

There doesn't seem to be anything that helps very much with those hard bumps except time. Moisture + SPF should keep them from getting worse, at least. Sigh.

your skin sounds similar to mine...
I have been using equal parts Peter Thomas Roth AHA/BHA gel (https://www.amazon.com/Peter-Thomas-Roth-Clearing-Ounce/dp/B00A9ELDS0) + plain old fashioned aloe vera gel for about a year & a half now, and it is just perfect.