Needle will not move

https://jukiquilting.com/media/catalog/product/d/x/dx-2000qvp_manual.pdf

Likely lint build up under throat/needle plate. See pg 118 for cleaning.

Top thread is "frayed"?

Could be tension is too high, problem with needle, needle wrong type, other.

Does it straight stitch ok using same fabric, thread, needle, bobbin, and tesnsion?

Generally, you would use a straight stitch. Trying a decorative stitch or zig-zag with a twin needle is tricky because the twin needles will hit the presser foot or needle plate if the stitch width is set too wide. Be sure you are using a presser foot with a wide opening.

Stitching Odyssey How to use your twin needle

How to Use Twin Needle on Your Sewing Machine

A twin needle makes neat pintucks if you have a pintucking foot.

I believe your machine is an older model with cogs inside. There is a connection to the needle arm. As the cog rotates it moves the arm in and out causing the needle arm to do the same. You have to check inside the machine to see if the connections are tight. As the connector moves the needle arm is pulled in constantly by a spring so the straight stitch is always to the closest side (the right). This may be confusing but I've sent a picture for you to look at. It has to be adjusted somewhere. Without seeing it I cannot give too much information. Anyway, I would like to know how you do.

Difficult to diagnose. Try a brand new needle that is compatible with the thread and fabric.
All About Needles

If the machine is mechanical, it may need cleaning and oiling if the decorative stitch mechanism is sticking.

...

The "dial" in the back is for decorative stitches which require a "programmer" to be inserted with the stitch formation you want to form. For a straight stitch no cam is required in the back of the machine. The three dials on the front:

Upper Left: Stitch selector
Upper Right: Stitch lenght
Lower: Stitch width

Using the upper left dial select the straight stitch picture
Using the upper right, choose your stitch lenght
Using the lower, choose your needle position, left/center/right

Hi Susie,

I can't tell whether you mean straight stitching and then turning your work 180 degrees to stitch atop the straight stitches in zig-zag, or continuing around the corner of your straight stitched side using zig-zag. But you can do either. You can mix up straight and zig-zag stitching at will.

Just ease up to a stop, ensuring the needle is positioned down inside the work when you stop.
Lift the foot.
Adjust to your desired stitch setting.
Lower the foot.
Turn the wheel toward you manually to ensure the stitches will be placed where you want them, and
Slowly engage the power again (via foot or thigh lever).

Hope this helps.

The machine may be slightly out of time so it works in straight stitch but not zig-zag.

First, make sure you select the straight stitch setting on the round dial. It is the one that has the dashed lines, looking like the stitch itself. Then, you may also want to be sure that the stitch WIDTH (dial with the picture that looks like a straight line gradually turning into a zig-zag) to the lowest setting, which is generally 0. With some machines, even after selecting a straight stitch setting, moving the stitch width can change needle position. Sometimes it will initiate the zig zag, other times it will move needle position left to right. Does that fix it?

Because it is almost a straight stitch but the needle continues to move in a zig zag fashion is reason to believe it is in a long stitch. By changing the number of stitches per inch there will be a very noticeable looking zig zag stitch.