Can you post a photo of your current setup?
By Shimano 'Switch' shifters do you mean 'bar end' shifters like these mounted on your aero bars?
I'm not sure why those would cause you to come to a complete stop as that's exactly what a lot of us run on time trial bikes but is that what you're talking about?
If so you've got a few options, you can remount bar end shifters from the ends of your aero bars to the ends of drop bars. Prior to shifters being built into brake lever assemblies that was a common way to get shifters up close to the handlebars. That just requires moving the shifters and wrapping the new longer shift cables and housings under the handlebar tape.
Or you can scrap those bar end shifters and mount integrated brake/shifters (a.k.a. brifters) in which case you just follow the installation instructions that come with the brifters when you buy them. Those will set you back quite a bit and you have to make sure they're compatible with your derailleurs. If you've got 8, 9 or 10 speeds in the rear then you want to match that to an 8, 9 or 10 speed set of brifters up front.
Swapping shifters isn't very difficult, but it will likely require new shifter cables and housings as the lengths will likely be different when going from aero bar ends to drop bar bar ends and you'll need to recable both shifters and brakes if you swap to integrated brakes/shifters.
BTW, you remove Shimano bar end shifters by first unscrewing the shift mechanism axles to remove the levers then you insert a 6mm allen wrench down the body of the bar end shifter to a hidden bolt that is left hand threaded (unscrews with a clockwise motion so backwards to most common bolts). Loosen the expansion bolt and the shift body slides out of the bars. This of course is much easier to do if you disconnect the shift cables from each derailleur first.
Good luck,
-Dave