Learn Farsi: من الان وقت دارم.

The verb dâshtan (to have) is a special exception in Farsi. While almost all present tense verbs start with the prefix mi-, the present tense of "to have" is just the root plus the personal ending: dâr + -am. It never takes the mi- prefix in its simple present form.

Since dâram already means "I have", do I need to include the word man (I) at the beginning?

No, you don't have to! Because the -am ending on dâram clearly tells the listener that "I" am the subject, you can easily drop man. Saying alân vaght dâram is perfectly natural and very common in conversation. Including man here just adds a little bit of emphasis.

Why is alân (now) placed right after the subject?

In Farsi, time words like alân usually appear very early in the sentence, typically right after the subject and before the object. The standard order is Subject + Time + Object + Verb (man + alân + vaght + dâram). You could also put it at the very beginning, but placing it before the object is the most common pattern.

Does vaght mean "time" on a clock, or "free time"?

In this sentence, vaght means having the time or availability to do something. If you wanted to ask about the time on a clock (like "What time is it?"), you would use the word instead.