Basically there are two things you can do to eliminate the two spot lights, you will probably want to do both - and I think both are not possible with the arm light attachments (so you may need to get other lights)

1. Change the angle of the lights

   The "spotlight effect" you see is actually the reflection of the light on the painting, if the light are at a close angle to the camera the reflection will be visible in the photo - but if the light is at a very different angle then the camera then most of the reflection will be projected "sideways" and not into the lens

   Note that putting the light at a very shallow angle will really exaggerate the texture of the paint and paper - so don't over do it.

2. Diffuse/soften the light

   To diffuse the light you really need to make the light source bigger, putting a white cloth on the light helps a little bit, putting a big cloth a small distance in front of the light will be much better.

   That is why photographers put umbrellas in front of flashes - it spreads the light so the the entire umbrella becomes a huge light source instead of a tiny flash bulb.

Now both of those will lower the usable power of your lights, by projecting the reflections away from the camera you are also throwing away a lot of light and spreading the light over a large area will significantly reduce the intensity of the light.

You either have to compensate with the camera settings or get more powerful lights (or possibly both), flashes are popular because they give out an enormous amount of light (for a very very short amount of time) giving you the ability to waste most of the light but to get better results.

Also, as R Hall said, color management can be a major issue but you can overcome it by using only the same controlled lights (close all windows and dim the room lights) by taking a picture of a gray card in the exact setup you use at the beginning of every session and by shooting in raw - then later you can use any raw processing software you want to calculate the light balance based on the gray card and apply to the rest of the images - and use only good full spectrum lights - no cheap fluorescent and led lights (modern higher end fluorescent aren't bad, indecent bulbs or good photography lights are best)