I agree that you need to see a doctor for the proper diagnosis since your pain could be caused by soft tissue pain (e.g. tendons, muscles, etc.), improper fitting shoes, the way you walk, etc. Shin splints is not actually a break in the bone, it is a term used for the syndrome. To actually find out what is going on you need to see a doctor.
In the meantime, when you get home from work, rest your legs, elevate them, trying icing them, and you can try an over the counter pain medication if needed. Let us know how you do. :-)
I do not think that you have any break in the shin bone, otherwise it would also hurt to put your weight on that leg.
It is surprising where you can get referred pain from. For example degeneration in your lumbar area can cause referred pain down the legs and into the feet.
I have degeneration in my knees and the pain is not felt in the knee, but all the way down the leg - more so when sitting and lying in bed, it too feels like it is the shin bone that is the problem.
The only way to get a proper diagnosis is to see your doctor and get an image of the lower back and whole leg.
Your profile does not say how old you are. Could you give us your age? Do you take any medications for arthritis or osteoporosis? Have you ever had any spontaneous bone fractures? When was your last bone density? Have you ever had shin splints? Have you been walking differently because of the discomfort in your knees? This does not sound like referred pain to me.
If you are not already seeing a rheumatologist, I would recommend it. There are many medications available for arthritis these days. However you could have another problem that this type of doctor could diagnose.
Have you tried elevating your legs and using ice or heat when you get home from work?
Your doctor has already said that you have arthritis in our knees and it if has been a while since he gave your this diagnosis, it would be worth your going back to the doctor and requesting an xray of the knee on the bad leg.
It is highly likely that the arthritis has got much worse and you may also no have osteophytes. The pain that you are feeling down the leg is referred pain.
Your doctor may advise you to take pain relief medication, prescribe you with an anti-inflammatory (take care with these as no everyone can take them and make sure that you take the anti-inflammatory with or after food because they can upset the stomach). Your doctor may also prescribe an antidepressant that is specifically used for referred pain.