My MS is also heat intolerant so I stay out of the midday sun, especially if exercising. Take your walk early morning or late afternoon. You can get cooling vests which hold ice packs, take a bottle of cold water (freeze it over night, then get it out before your walk, let it defrost so you still have ice in the water). You can walk around an air conditioned shopping centre.

Ditto to AndreaG. Stay out of the sun as much as possible, because it'll only make things worse. The only suggestions I can add are wearing a floppy hat and possibly tying a wet rag around your neck. The drawbacks to that are that it will soak the collar of your shirt and dry out quickly. They also make little fans that you can wear around your neck, but I don't know if they actually work. I have a cooling vest, but I don't use it often because I'm fat and it's awkward to wear.