Started a Part Time Job. Would it be wise to apply for other ones?

After two weeks of job hunting, I got a call last week from Family Dollar and was offered a Customer Service Representative position. The ads said it was a full time position, but I found out that the position actually given to me was part time. My question is, if I were to get another call to hire me as a full time worker, should I go for it and leave Family Dollar? If so, how do I address this to my manager considering I have barely started.

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