The goal is to edit a multi-camera shoot to a single video with a unified audio track. The options abound.
As noted above NLEs can do the sync in post. All you need is some common, event in the video, such as a clapper board, flash or rim shot, or other clear short duration event in each video to visually sync them up. Some NLEs support multi camera editing - you can see all synced cameras (three in your case) playing on one screen and click on the one you want to edit quickly. More sophisticated systems will provide house sync and time to all cameras, but that sounds a bit beyond your needs.
If it is all in one venue, you can start/stop the camcorders at the same time using a remote control provided they can be set to recognize it. There are also wireless systems available for extending the range of camcorder and VCR IR remote controls. This should give start/stop accurate to within a frame or so.
A couple examples follow (but I have not used them nor have I used your cameras).
https://www.amazon.com/Wireless-Remo...a-566982835846
https://www.mycablemart.com/store/ca...yABEgIHKPD_BwE
You need the "good" sound on one camcorder. It can be used as the master audio track in your edit. Ambient sound is OK on the others because you can select you sound you use in the NLE. Some folks will record the "house" sound system to a portable recorder such as a TASCAM DR-40, not to the camcorder, for sync later in the NLE. This can save managing/running cables and give better sound. Any you can use wireless mic systems to get audio to distant camcorders as well, but that requires appropriate selection of gear to allow one transmitter to feed multiple receivers, and coordination with the venue to avoid any conflict with wireless gear they might be using.