Let's start off with running shoes versus walking shoes. Yes, there are shoes made specifically for walking just as there are made for running. That being said, I strongly recommend that you have a pair of shoes for running and a different pair of shoes for walking. Though running shoes can be used for walking, they are not made specifically for walking.
Running shoes and walking shoe have different designs, since running and walking are two different motions. Running shoes usually have more cushioning in the heel while walking shoes have more arch support. Walking shoes tend to be more flexible in the forefoot area rather than the midfoot area of a running shoe. There are some running shoes that work well as walking shoes, but your best bet would be to get a pair of walking shoes since that is what you do the most.
Here at RunnerClick, we do have a list of the Best Walking Shoes which you will find here. These shoes will provide most people with the right amount of cushioning, flexibility and support needed for walking. Some of the shoes on the list, including the Saucony Grid Omni, come in wide width too. However, the only way to find the best fitting shoe would be to try it on, walk around the store and see if it feel comfortable for you.
Also something to keep in mind with walking shoes, as well as running shoes, is to replace them regularly. Running shoes will usually last 300-500 miles. For walking shoes, if you walk 4 times a week for about 45 minutes to an hour, you probably need to replace your walking shoes every four months. If you walk more than that, you will need to replace them more often. Wearing shoes that are worn out can lead to injuries (or make the ones you are currently experiencing worse).
With all that said, there is no "one size fits all” approach to shoes. And considering all the issues you are currently experiencing, I would strongly suggest that you see a podiatrist and/or primary care physician as soon as possible. While finding the proper shoe is very important, if you are already experiencing issues such as plantar fasciitis, simply finding the "right" shoe will not cure those issues.
In the meantime, we do have an extensive injury section that has a lot of information about the issues you mentioned in your question. There is a plantar fasciitis guide as well as a shin splints guide, both of which provide information on causes, treatment and prevention.
But as I said before, I strongly suggest you go see your doctor first to treat your current issues as well as figure out the best walking shoe for you.