If you play be ear, this will probably be a lot easier than if you read and/or think in terms of note names.
1. Put your other basses in their cases and leave them there until you get really comfortable on the 6. I believe most people will progress faster and have less frustration if they do this. Picking up old faithful tends to reinforce the old way of thinking...and it can come back with a vengeance.
2. Start off slow and only play the strings you are used to. The idea is to get used to the how the extra string impacts your perception of the strings you are used to.
3. Add the C string a little at a time. Study and practice to learn and reinforce the expanded limits of your new fretboard.
4. Work specifically on muting. You will need to use both hands and floating thumb is probably going to be a must.
The first goal is to elevate you playing on the 6 to the level you were at previously on the 5. If you play daily, you should get comfortable fairly quickly (a few months), but you will probably continue to have lapses where you get lost (what string am I on) or have problems with strings ringing unintentionally. It may take a year or even two before you really get confident and have developed and mastered the expanded muting techniques you will require.
Hopefully mastering the 6 will eventually allow you to take your playing to the next level. Over time having the extra string will change the way you think about navigating the fretboard. On a 4-string, you tend to play up and down the neck more. With a 6-string you often play across the neck, which is more fluid and efficient. Also with 6-string you may choose the range of the neck you want to use depending on the tone you want. I would say it took me a about a decade to really start thinking like a naturalized 6-string player, rather than a 4-string player with a few extra strings.
When I transitioned from 4 to 5, I continued using essentially the same muting techniques. when I transitioned from 5 to 6, I found it necessary to develop new muting techniques. Now I rarely think about muting; it just happens automatically. But occasionally I play something that requires me to pay conscious attention to muting. For example this year I have been working on playing the melody of Groovin High up around the 15th fret, and sometimes I notice the B string ringing.
I still own 4s and 5s. I pull them out and play them once or twice a year. If I play one of the them too long, it starts changing the way I think about navigating the fretboard...so in effect it degrades how well I play on the 6-string. So even after almost 20 years with the 6 being #1, for the most part I still avoid playing 4s and 5s.
I do not have any problems switching back and fourth between my 6-string bass guitar and upright bass or electric upright bass. I use different fingering system on upright (Simandl) and I started using OFPF on bass guitar shortly after transitioning to 6-string. So I use totally different fingering systems on BG VS UB.
