Damon, your F 350 should not be shifted without the clutch because the synchronization cannot work with the clutch engaged. All one is doing by shifting it without the clutch is wearing the synchros.
I know that Eaton now has some auto-shift 18s out, but I know nothing about them.
On our 18 speeds that are older, I double clutch sometimes, depending on the circumstances, but float the gears in the majority of the time. This is on an older mechanical Cat that the throttle can be adjusted slightly with ease as opposed to many of the newer engines that have much snappier throttle response, in which case shifting without the clutch becomes primarily timing, and misses are harder on the drivetrain.
I second Ron G regarding double clutching. It is the best way to get familiar with the trucks attributes without casing damage. Eventually you will find yourself just barely touching the clutch when entering a new gear, but until then it is best to double clutch for the sake of the equipment.
In case you do not understand the purpose of double clutching, it allows one to take the transmission out of the gear that it was in and into neutral on the first clutch, then you match the rpm of the next gear that you are going to with the clutch engaged (in neutral), then you put the transmission into the next gear with the clutch disengaged to remove shock and allow for minor adjustments of the trans/engine rpm match the next gear requirements. The reason that you need to engage the clutch in neutral is that you cannot slow down the trans (or speed up if downshifting) with the clutch disengaged.
Also, do NOT depress the clutch clear to the floor when shifting as this will activate the clutch brake and wear it prematurely. The clutch brake should, with few exceptions, be used only when stopped and wanting to stop the spinning of the clutch in order to place it in gear. In addition, never use the clutch brake to hold the truck on an incline while waiting for a light to turn green as it is not designed for this.
I hope that this makes sense, but feel free to ask for clarification. Others may be able to describe the functions better than I have.