I would say つめたい usually refers to objects (including body parts), whereas さむい is more commonly used for the general temperature. What I wouldn't say though, is わたしはつめたい as it sounds a bit... odd? (Probably because つめたい is also the word used to descibe a cold personality.) But if it's your hands or nose or whatever that's cold, tsumetai works perfectly. Just saying "it's cold" (さむい) is enough to let people know that you're feeling a bit nippy.

When the object itself is cold, we use つめたい. Ex.) みずは (or みずが)つめたい。(The water is cold.) かぜは (or かぜが)つめたい。(The wind is cold.) When the person oneself feel cold, we use さむい. I'm cold /I feel cold. (わたしはさむい。) In Japan, winter is a cold season/it's cold in winter. (にほんではふゆはさむいきせつです。) *As Klara-san said, わたしはつめたい means "I am a cold person/my personality is cold", so you need to care about that. I hope you understand a bit more.

When you touch something, and that thing feels cold. You should use つめたい I'm cold - （わたしは）　さむいです。