Yes, it is considered impure until one takes a bath; performance pratha sandhya is not considered a purification ritual, but taking a bath is crucial.
Notes:
There are differences between theetu and vizhuppu.
Vizhuppu means not to touch others or anything, removal of old clothes that were worn the previous day after waking up; the cloth that was worn the previous day is known as vizhuppu.
Theetu is more than vizhuppu; at times it happens like coming into contact with someone or something.