It’s normal to feel awkward when insisting to use your own cup. But don’t let that stop you.
I have been using my own Klean Kanteen #reusable steel pint glass and porcelain mugs in lieu of single use paper and plastic cups for years at coffee shops, but have always been a little leery of using them at quick service restaurants. Some, like McDonald’s and other places you shouldn’t be eating at anyway, freak out when they see you use your own cup. Most franchises and mom and pop shops, I have found, are actually happy when you use your own cup, as it saves them money.