Shin Splints & Fallen Arches

See a podiatrist.

You didn't mention how far you're walking & the type of terrain etc but shin splints (to some degree) are not that unusual with changes in activity level or switching from a treadmill to running outside. Rest & ice will normally look after them, dial back the activity level a bit and they should go away fairly quickly.

The foot pain is more concerning....do you actually have flat feet or did the shop make that diagnosis?

Here's a DIY test you can do to check for yourself...

runnersworld.com/running-shoes/the-wet-test

It sounds like the shop may have sold you an inappropriate shoe, can you take them back and exchange them?

If the foot pain doesn't subside it may mean a trip to the podiatrist.

My experience with shin splints was mainly during high school when I ran track. My coach had me stretch for at least 30 minutes and my shins were iced at the same time. There was basically a wooden box with one side at a 45 degree angle...so it looked like someone had sliced the box diagonally. I would put that box against a wall and would place my feet on the angled portion and stretch. It really stretched my calves and the ice helped the pain.

Stretching also helped when I used to wear heels and my arches were dying at the end of the day. (Arch very sore and toes cramping up which caused them to curl down.) I didn't have that wooden box so I'd just sit on the floor, with my legs straight and in front and bend at the waist, trying to touch my toes. Or I'd have one leg out straight, the other bent like your sitting criss-cross and try to touch my fingers to the toes of the leg that was straight. Some days it'd take 20-30 minutes for the arches to relax. I don't wear heels that often anymore because it's just not worth it.

And finally, you may want to consider changing the way you walk. Since I have high arches, I've read I should try to strike the ground with the front of my foot instead of heel first. This is hard to do since the majority of us learned how to walk doing heel strike. You may be more tired at first since you're working leg muscles in a new way. I've tried changing the way I run and while I still need to practice running that way, it's helped lessen the pain and I can run for longer periods. (Still get achy knees and arches before I get tired everywhere else but it's an improvement.) I also got shoes that are designed for running and for people with high arches and that definitely helped. Are you wearing shoes designed for walking and for your particular foot/gait?