My mom is in assisted living, trying to figure out how to deal with her incontinence.

You are not going to have a choice, in most ALF, to add this service. This service will be a REQUIREMENT once the staff realizes the added problems involved here. This requires more staff time and the level of care (and consequent cost) will be increased. ALF all have levels of care, usually one through four. And the cost goes up according to the needs/condition of residents. Purewick is complicated and anything but foolproof requiring positioning at night in bed, and is for bedbound people. Some doctors believe they increase infection. You can read all about the device on their site. You can also access videos on youtube by simply looking up "youtube tutorial use of purewick catheter" and will get films to watch.

Again, this is a problem that will increase costs. Sorry for that, but you are already seeing the results of inadequate staffing, and increased problems means increased staff needs. It's a dilemma.

If they cannot afford to pay for incontinence care in Assisted Living, then your parents cannot afford Assisted Living. Care costs only increase as they age, so bear that in mind moving forward. Your stepdad did not move into AL to become moms caregiver and to be doing loads of urine soaked laundry every day, let's face it.

Short staffed or not, the AL will have a responsibility to deal with moms incontinence care entirely once it's paid for and part of her care plan.

Pay the extra amount of money to have an aide do the changing. Or, ask that one of the aides train Dad in changing diapers. There is a nack and I knew nothing about it till Mom was in AL. First, pull ups tear down the sides. Second, diapers can be put on without taking off outer pants. You pull the outer pants down to the ankles. Put one side of the diaper around an ankle and slip it up one pant leg over the crouch of the pants and hook the other leg of the diaper over the other ankle. Then u can pull both diaper and pants up at the same time. Your Dad should be taking Mom to the toilet ever two hrs.

Pay for the extra service. It will be worth every penny.

Some premium overnight briefs (for example, Abeena Level 3) hold an astonishing quantity of liquid and significantly reduce the laundry load. 

These can be dropped into an odor-sealing plastic diaper bag and into a metal can. Then into the garbage it goes. 

Maybe worth a try?

I cannot imagine why they wouldn't allow Purewick. The home version is just a dinky little plastic thing. It plugs in like (and sort of looks like) a bedside humidifier. It's not like the hospital version. Cable news runs Purewick commercials night and day. It's just another overpriced home consumer item.

You would have to ask the AL if they allow the Pure Wick.
(and there is someone that just posted that they are trying to sell one.)
They are not covered though.
But...
Bottom line is that your mom is having more problems than you or your step dad can handle.
Either paying more for the service that is needed or having someone come in and take the laundry out to be done. That someone could be a family member or a caregiver that you pay to manage laundry. And if that is the route the caregiver can do other tasks that you may be doing.
Incontinence will not "get better" she will become fully dependent on the disposable briefs and the rest of the laundry clothing, bedding will need to be done more often and and furniture will need to be protected.