If the travelers travel through the ceiling boxes, then it's do-able, and you won't need to pull additional cable. Start by opening the ceiling boxes to make sure you have two travelers in both boxes and identify which fixture has a single switchleg connection (and which fixture box has the switchleg traveling though it). It is also common to have the Home-Run power/feeder in the ceiling.
LET'S FORGET ABOUT THE HOME-RUN HOT IN THE CEILING:
Mark the switch with the hot/power as Switch 1. Mark the other switch (with the switchleg) as Switch 2. Identify the travelers A and B.
Switch 1: Disconnect the Hot and Traveler B. Connect the Hot, Traveler B, and a pigtail together with a wirenut. Connect the pigtail to the common screw.
Switch 2: Disconnect Traveler A and cover the end with a wirenut. Exchange Traveler B with the switchleg (in other words, move Traveler B to the common screw terminal and move the switch leg to the traveler terminal).
Now, you need to decide which switch you want to control which fixture. Call Fix 1 the fixture that Switch 1 will control and Fix 2 the fixture that Switch 2 will control.
Identify the ceiling box with "two" switchlegs that connect both fixures; in this ceiling box, there will be two wires connected to the fixture-power-supply via a pigtail, or you may also find two wires connected directly to the fixture-power-supply wire. Disconnect the two switchlegs from the fixture, and identify the side connected to Switch 2. The other side of the leg is connected to the other ceiling fixture. Mark the leg attached to Switch 2 as Leg 2. Mark the other leg as, "Fix Leg"
Identify Travellers A and B. Traveler B is now the Hot for Switch 2. Traveler A will be the switchleg for Switch 1. Disconnect Traveler A. Mark the side connected to Switch 1 as Leg 1, and put a wire nut on the side of Traveller A that is going to switchBOX 2. This side of Traveler A is not being used anymore. Personally I would mark that wire as "dead" or "chicken" or "banana".
Now, it's time to employ your descision making skill. Is this ceiling box for Fixture 1 or Fixture 2?
If you are working in Fix 1, connect Leg 2 and Fix Leg together (so that Switch 2 is powering Fix 2).
OR: If you decided that this ceiling box belongs to Fix 2, then connect Leg 2 to this fixure (and don't connect Leg 2 to Fix Leg).
Now connect Leg 1 (from Switch 1) to either this fixture or Fix Leg (whichever one is left over from the previous step).
Double check your work before closing the boxes: Traveler B is now marked hot for Switch 2. The switchleg for Switch 2 was moved to a traveler terminal and should be identified as, Leg 2. Leg 2 is connected to Fix 2- does it travel through the ceiling box of Fix 1? Traveler A from Switch 1 has been marked Leg 1. Leg 1 is connected to Fix 1. Does Leg 1 travel through the box for Fix 2? The remainder of Traveler A (from which ceiling box to Switchbox 2?) is no longer in use and both ends have a wire nut.
IF YOU HAVE THE HOME-RUN HOT/FEEDER IN THE CEILING:
There are several ways to accomplish rewiring in a more expedient manner (than described in the previous directions). However, describing the intention is perhaps the simplest direction.
Each switch will need power supplied to the common terminal.
Each switch only needs one leg to supply power to one fixture.
The fixtures should not be connected, but since they are on the same circuit they may share a neutral.
Best Answer
First, don’t experiment i.e. trying random stuff “to see if that fixes it”. There are lots of combinations that will work and will kill you.
Second, you are quite close on identifying the wire’s operational roles. They are:
If the first and second are connected, then the rest of your house will work. And they should always be connected no matter what you do.
What’s driving you up the wall is that you are continuing a sloppy practice someone else started, which is using the switch itself as a junction block for unrelated things. (I.e. the connection between supply and onward).
Very few switches even have provision for 2 wires on the line screw; that generally forces you into using a “backstab”, which is a bad idea all around.
You may be better off obtaining wire, nuts and the other kit required to make this a pigtail connection: join the two important wires (supply and onward) under a wire nut, so you are not using the switch as a splice, and then have a third short wire hop from the splice to the switch. This will also make the circuit more understandable for the next person. (Who might be you!)
I pigtail most of my stuff for a different reason, most of my connections are in awkward locations. It is easier to pigtail the outlet on a bench, then at the box simply whip together supply, onward and pigtail with a wire nut.