Pembroke Welsh Corgis do have some health issues . The PWCCA suggests testing eyes and hips (OFA or PennHIP). 

The additional testing I like to see done are the following:

vWD1 (Von Willebrand's Disease Type I)
Von Willebrand disease (vWD) is an inherited bleeding disorder that inhibits the normal blood clotting process, causing prolonged bleeding after an injury. People with this condition often experience excessive bruising, prolonged nosebleeds, and excessive bleeding following any form of trauma, including surgery, or dental work.
The primary function of von Willebrand factor (vWF) a blood glycoprotein, is to bind itself to other proteins. The deficiency or failure of vWF function inhibits the blood coagulation process and causes bleeding which is most apparent in tissues having high blood flow or narrow vessels.

In dogs (as well as in people), three separate types of vWD have been identified. Of these three types there are five different genetic mutations causing vWDs in dogs. Genetic tests have been developed to identify all five variants. Von Willebrand's disease type I (VWDI) is transmitted as an autosomal recessive trait with variable penetrance and is the least serious of the three.

The causal mutation for VWDI is a G>A substitution. The mutation reduces the normal production of von Willebrand's factor (vWf) in affected dogs to only 5-10% of normal vWf production. This low concentration of normal vWf prevents normal blood coagulation resulting in excessive bleeding.

Requires both parents to have or carry the gene for vWD1. vWD1 is a recessive disorder and a dog must have two copies of this mutation for the disease to manifest. This means that a dog can have one copy of the mutation and not experience any signs or symptoms of vWD1; this dog would be known as a carrier. The carrier can then pass on either the normal gene or the mutated gene to any offspring. If two carriers are bred, a dog could potentially receive the mutated gene from each parent and be affected by vWD1.

Symptoms include: Abnormal Bleeding, Abnormal Blood Clotting, Bleeding Gums, Blood In The Feces, Blood In The Urine, Easy Bruising, Excessive Bleeding, Hematochezia, Hematoma, Hematuria, Nosebleeds, Vaginal Bleeding.
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EIC (Exercise-induced collapse)
EIC is characterized by muscle weakness, lack of coordination, and life-threatening collapse after intense exercise in otherwise apparently healthy dogs. Affected dogs tolerate mild to moderate activity but will display signs of EIC after 5-20 minutes of strenuous exercise. The severity of EIC varies. EIC episodes last from 5-25 minutes with a gradual return to normal with no apparent residual weakness or stiffness.

What about DM?

Let's discuss DM! First, this is what it is... 

DM is a POLYGENETIC disease.  Let me repeat this... POLY (meaning more than 1).  

Degenerative Myelopathy (DM) is a progressive neurological disorder that affects the spinal cord of dogs. Dogs that have inherited two defective copies will experience a breakdown of the cells responsible for sending and receiving signals from the brain, resulting in neurological symptoms. The disease often begins with an unsteady gait, and the dog may wobble when they attempt to walk. As the disease progresses, the dog's hind legs will weaken and eventually the dog will be unable to walk at all. Degenerative Myelopathy moves up the body, so if the disease is allowed to progress, the dog will eventually be unable to hold his bladder and will lose normal function in its front legs. Fortunately, there is no direct pain associated with Degenerative Myelopathy. The onset of Degenerative Myelopathy generally occurs later in life starting at an average age of about 10 years. 
Symptoms include: 
Hind leg weakness and eventual inability to walk. As disease progresses dog will lack bladder control and will lose normal function in front limbs.
**Disease itself is NOT painful, and is onset around 10 years of age +
**Disease CANNOT be diagnosed without a necropsy - therefore only after the death of the dog. There are many things that mimic DM that are not DM.

But there's a DM test! No dog should have DM if you can breed it out! RIGHT?  Mmmm no. Sorry! It's a marketing ploy!
You'd think yeah? No.  The DM test is based on 1 gene. The SOD1 mutation. That's a single gene that so far we do not even know how it affects DM in a dog. It could be what keeps it at bay until the dog is older. It could mean absolutely nothing, or everything. The point is... we don't know what this single mutation on a gene on a polygenetic disease. Top geneticists stress the importance of breeding from a genetically improved pool of dogs across the board rather than removing a HUGE amount of animals from the breeding pool based on a fallacy built on getting labs money.  Science is wonderful.

Then why is there a DM test?
Again, it's not a DM test. You're falling for a marketing ploy. It's an SOD1 test. That's it. It doesn't mean your dog will or will not get DM as there have been too many dogs necropsied as not having the disease but having the SOD1 mutation. The actual evidence from UC DAVIS and MSU indicate that the amount of dogs actually dying from DM is to be .06% in the ONLY study ever done on it. That works out to be about 3% of all Pembrokes born will have this affect them.  

Your dog has a greater chance of getting hit by a car, drowning in a bath, having cancer, heart failure or spinal issues non-related and dying from those things than actual DM.