It seems very strange to me that there are no prices on the menu. Are you sure about that? I've been to restaurants that organize the cocktail menu by price. There might be a $6 page, an $8 page, and a $10 page, for instance. Sometimes it's hard to find the prices because they aren't right next to the menu items and they may be written in a more subtle font.
Anyway, I understand your hesitation about asking, though I don't think it would be rude to do so. MariaE has a good idea if you decide that you'd prefer to not ask while you are sitting with your relatives.
Are your relatives treating? If so, and they're ordering cocktails, you probably have a green light to do the same. You can ask your hosts for a recommendation (and assume that they will suggest items that are within their "acceptable expenditure" range), order what they're having, or order a well drink, which is likely to be cheaper.