Most perennial groundcovers, if selected with site conditions in mind, will not need much supplemental water during the growing season. Site conditions include how much sun exposure (in summer) the area receives, how well the soil drains, and if any areas stay drier (due to soil type or an overhang) or wetter (due to drainage patterns or low spots, as you mention). Whether or not deer browse in the yard is another important consideration; some groundcovers are much more unpalatable to deer than others.

If the area receives at least 6 hours of direct sun daily in summer, then it is considered "full sun". If not, it's "partial sun" to "shade" (if it receives 3 or fewer hours of direct sun). Most groundcovers/perennials fit into either category; not many can tolerate both. That said, you don't have to have a single type of groundcover throughout the area; choose what's best for any particular portion of the yard and use a mix of species overall. That means that your lower-lying damp spot can be planted with a different species than other areas. Similarly, if you have some areas that are shadier than others, use different species in each. They will mingle where their tolerances overlap, but otherwise thrive in separate conditions, making the planting overall as low-maintenance as possible.

We can't make specific plant suggestions without more information about the site conditions. A photo may be helpful, yes, if it shows the overall part of the yard of concern; you can attach the image file to your reply.

Regarding the dog, though, no groundcover stands-up to dog digging very well. Turf, actually, tends to be the best for this, which is why it's popular when people need a relatively dog-proof part of the yard (and why dog park plantings are largely turfgrass). Even so, the root disturbance and soil compaction of frequent use will inhibit the growth of any plants in that area. Perhaps a different ground-covering material would be the most practical for a particular part of the yard that the dog uses most, such as artificial turf or pet-safe mulch. (Consult with a vet if needed.)

Several evergreen groundcover options are given in our groundcover list page. It's not an exhaustive list; you may need to visit local nurseries to see what species they offer to expand or narrow-down potential choices. Native species are preferable for wildlife, but few of our native groundcovers are evergreen. At the very least, do not use invasive species like Vinca minor, English Ivy, Japanese Pachysandra, or Ajuga.

The vast majority of perennials and groundcovers sold by area nurseries will be reliably winter-hardy. All go dormant in winter (in that they don't actively grow above-ground) but some are evergreen (keep their leaves in winter). Overall, more species of groundcovers are deciduous (losing leaves in winter) than evergreen. Not removing the dead foliage in winter, though, will allow it to protect the soil from erosion and weed seeds as a natural mulch.

Affordable is difficult to quantify. For some native plants, nurseries offer smaller pot sizes called "plugs" - they are younger plants than what is typically sold, and sometimes sold in greater quantities (not singly) to reduce the price even further. They will be less expensive, but will take longer to establish and fully cover the soil. How quickly any of them cover ground depends on what species they are and how closely they are spaced apart upon planting. The least-expensive approach would be to sow seed, but fewer species are available this way, and here too, this extends  further how long it will take for the planting to fully cover the ground. Plus, with starting from seed, it may be harder to recognize weed seedlings as they germinate to know what to remove. You will need to decide if any of these trade-offs in speed are worth the reduction of expense; there is no right or wrong way to do it, just what works best for you.

Miri