Indicator Lamp (LED) on 3-Way Switch
1: You are replacing switches and outlets room by room .

2: Since you said 3 way and two switches there - did you replace those switches with 3-ways and do you know how to correctly wire up a 3 - way switch  ?



EDIT 11/14/2017
 The light I assume was already in the circuit and had two wires to it. Your hot and neutral that go out to the light in the garage is what the LED would connect to just like the diagram above (parallel); try hot on one lead neutral on the other swap the wires around if it does not work. The previous owner might have wired it in series with your garage light- this approach may not work properly with an LED which is a Diode (that emits light) and passes DC only, thereby reducing the voltage to the garage lamp because you only have half the wave of the AC. 

Given your description states Red, Black and White and that the led flickers  - sounds like he ran the light in Series with the LED You are half waving - half the time the LED lights (positive cycles / negative cycles) - half time on half time off 30 times a second it is flashing. 

EDIT 11/18/2017

To verify - turn off your Circuit Breaker - verify the circuit is dead - use a volt meter and also a circuit on/off test.

Step 1:
Disconnect your black on the left side of the switch and the red from the right side of switch. Now being careful not to touch those wires to anything - hands included - turn the CB on and test for voltage at the black wire to the switch. If Voltage is present - flip your other switch - just to verify and check for power. You should have power - this means you have switch #1 per the diagram (should be) - your neutrals - the white wires are tied together in this box (hopefully they truly are neutrals). 

If step 1 is correct:
Step 2
Check measure voltage on your black and red wires that you disconnected - DON'T touch the metal of your meter probes or any metal with your fingers/hands - measure the voltage from black to white (or copper) and also from the red to white (or copper )- this black and red share the same wire jacket. If voltage is present something is not correct. Your travelers are powered somehow on the other side - this should not be the case given step 1 and your picture.

If no voltage on Black and Red - they should be the wires going to your other switch (the traveler) - given your lights work with the LED removed - your second switch most likely has two sets of wires (red, black , white and copper) - OR there is something tied together that we are not seeing in your indicator wiring box. 
So at this point we need to have a picture of the indicator wiring box and the second switch.

Step 3:
If all is well so far - in your second switch you should have one set of wires - the traveler and the indicator wires. 
With your CB off verify what wires are which with an ohm meter. Measure continuity of black and red to your indicator .. if need be as long as the cb is off and you have no power - you can tie the black and red in your indicator wires together and measure at the second switch between black and red and verify continuity - this tells you what jacket goes where. (you could tie the black and red together - the ones that you disconnected from switch 1 earlier - and do the same thing to determine which set of wires is from where.

Now when you know which one is the lamp - you should have black and red as hot in any on position - so in your wiring that goes to the indicator - just leave the red wire nutted off on both ends. Connect your indicator jackets black lead (going to the indicator) to the common of switch 2 (per diagram below I have updated) and the white lead connect it to the neutral (at this point it should work) - so reconnect switch #1 according to your picture you sent the way it was originally (black on left and red on right). I usually take pictures for projects - because interruptions - I will forget how it was and I am too lazy to draw or write it down.

Turn your CB back on and test it. 

EDIT 11/19/2017

As I am looking at the three pictures I find the one that is bothering me the most is the indicator connection. I see a R,B,W,C wire set (which means it has to come from either outside of this circuit (switch 1 and switch 2) or it is 'spliced' / wire nutted in somewhere along the path of the traveler wire set (R,B,W,C) from switch 1 and switch 2. and you said when switches on that white was hot too (wrong thing by code). 



TO MAKE IT WORK

So if all are hot when switches are on - to make it work you can pick that wire nutted white wire (with the pretty red wire nut) back of box on the right hand side where the 2 two line sets are nutted together (verify that it is a neutral measure between it and the Red or Black of the (R,B,W,C) wire set; If you read 120V use that for one lead to your LED and then use the Black from the (R,B,W,C) wire set for the other - it should work. 

That said I would not say the wiring is correct as the (R,B,W,C) set in the indicator box - where does it come from and how is it connected ?