Unfortunately this seems to be a challenging situation. It is too late to overseed, as waning daylight (a major factor even if temperatures were mild) and upcoming frosts/freezes could damage seedlings, interfere with germination, and halt growth. You might be able to source some sod this late in the year, though even establishing that might be risky to have it root-in enough to withstand winter weather. Given the ongoing drought, watering will be crucial, but even aside from that trend, seed and sod cannot dry out for long while establishing its first few weeks after sowing/laying. If we start to freeze overnight, a frozen garden hose may make it harder to keep them regularly watered.

Partial shade isn't very conducive to thriving turfgrass, though the most shade-tolerant species are tall fescue and (even more so) fine fescue. However, fine fescue is the least tolerant of regular or heavy foot (human or pet) traffic. Tall fescue has average tolerance, and while zoysia is better, it still has its limits and even a vigorous zoysia lawn might struggle or develop bare areas for a time after heavy dog use. Zoysia is not shade-tolerant, which unfortunately won't help its ability to recover from damage. Seedling grass needs to reach mowing height at least once  before it should be walked on, so dogs and people would have to be kept off of it for several weeks, even setting aside the current timing challenges and the slower growth that would result from too little daylight and chillier weather (eventually). Sod has a head start, but should still be left alone and not walked or run on, ideally, until well-rooted so it can put up with being run over.

While plenty of alternative groundcovers are more sustainable than lawn, provide more ecosystem support, and can be more aesthetically attractive throughout the seasons, alas, they are not nearly as traffic-tolerant as turfgrass. One option for human traffic is to keep walking paths or play areas as turf or mulched walkways and to change-over other lawn areas to non-lawn plantings, reducing the area of lawn overall. That approach might not work as well for dogs, though, if they are prone to running into garden beds or other non-lawn plantings, especially low growers like many groundcovers.

Some gardeners opt to use artificial turf in areas with heavy dog use, but we acknowledge that more study probably needs to be done about microplastic exposure risk from its wear-and-tear or weathering, plus other health impacts.

If you want to try establishing new lawn in these areas for now, the multiple web pages within the Lawn Care and Maintenance group provide guidance. Soil aeration (core aeration) is definitely a good first step when you are ready to seed or lay sod, as it will help to alleviate some soil compaction. Soil testing (linked within the page group above) can also inform you as to what amendments (if any) may help turf to establish, and what kind of fertilizer can be used to avoid over-applying phosphorus.

If you decide to hold off on seeding/sodding until spring, then you could try laying straw or some other type of mulch to reduce the risk of erosion and weed seed invasion in the meantime. You might be able to find a product that has straw woven into a sort of "blanket" roll-out mulch, though sometimes these come with seed mixed in, which wouldn't be the best choice as that seed type might not be a lawn grass type you want to work with. Plus, some blankets might not use a biodegradable material for a plastic-type mesh that is used to hold the materials together, but there may be numerous products on the market that do different things, and this is just a generalization of what construction companies tend to use, so it might not apply to all products.

Feel free to ask for clarification on any of the lawn care/creation information on those linked pages.

Miri