What is the meaning of when someone says I want to run some errands.Does it sound a bit odd?

It means they have a number of tasks to do, usually at multiple locations away from home.

It's used in some regions of the English speaking world, and not in others. So it might sound 'odd' to you.

Depends where you are from, I don't think it sounds odd. It is a bit evasive, as they want to tell you they are busy, but don't want to tell you what they are doing. Which seems fair enough.

I want to run some errands.

(Meaning) :-

In this question, errand is a noun that means a short trip taken in the performance of a necessary task. An errand is a short trip to fulfill some small business. When you're running errands, you're making a bunch of these all at once. The noun errand can refer to the mission of the trip itself.

run an errand
Go to perform a commission, as in I spent the morning running household errands—to the cleaners, the supermarket, the hardware store. [c. 1500 ]

http://www.dictionary.com/browse/run-an-errand

It means I have things to do.

no, it doesn't sound odd. it means that someone has things to do maybe shopping. visiting a friend, seeing the doctor etc.

No, this is very normal. It means they have things to go do.

It has been used in Scotland for hundreds of years, so no it is not odd

Not really odd. People do have to leave home and take care of some shopping or banking matters.

I imagine it is a polite way to be dismissive. I want to run some errands, so I have to go now, or you have to go now so I can do what I want to do and had planned on doing before you stole me away from that.