There is no distinction between dependent and independent pronouns. I and my alternate as 1st person subject pronouns. Such alternation is more obvious with verbs like 'want' and 'think'. In fixed expressions such as 'I don't know', I is always used.
Source: Tong 1862: VI.1
Source: Tong 1862: VI.54
Source: Downing 1838: I.284