2003 Dodge Neon What is a coolant flush?

Coolant flush has almost nothing to do with your engine shaking. Look at the color of your coolant. If it is orange-ish, you are good for up to 100K miles, if it green-ish, then you should flush.

In a nut shell, coolant flush is basically removing MOST of your coolant/antifreeze from the radiator, the engine block, the heater core and the lines, and replacing it with new fluid.

I am more concern with your shaking problem. Can you give a little more descriptive explanation to the problem. Include time of day, idle or driving, intermittent or all the time, etc. The more description you provide, the easier the diagnosis will be.

I am going to assume that the shaking you mention is the engine lopes while idling or idling erratically.

A few things for you to check.
1. Vacuum leak. Check all your vacuum lines, making sure that they are all connected, no frayed or cracked lines. Here is your vacuum diagram

2. Leaking EGR valve
3. Clogged air filter
4. Insufficient fuel pressure

There are other tests that you should do, but let's start with these first. These are the most common.