I too use Stage mode for projecting presentation mode style lyrics - for others using opensongapp , turning off chords display in 'Connected Display' to get this.
My 2 button pedal is configured for short press scroll down / scroll next, long press next song / prev song.
I have pedal options
'Toggle scroll before Next/Previous song - ON
'Show a warning before moving to the next song...' ON
The 'Toggle scroll' options makes short or (accidental) long press in the body of a song move a section (very safe no accidental move to another song!)
To change song a long press at the bottom gives a warning, a second long press a little later then changes song. (very safe no accidental move to another song!)
I also have sticky notes displayed as a section at the bottom (a sticky notes option) - if you move to this section you get 'no lyrics' displayed at the end of a song.
I use 'Presentation' order to get songs top to bottom in the order needed (bottom is then always the end) - if it is a 'special' order I make a song variation for that set.
For the 'connected display' I set a short slide transition time of 0.2s.
On the tablet I can see the previous, current and next sections. With a little practice I can now change to the next section ahead of 'playing need' in in order to have it displayed for 'presentation need' - people need the words ahead of time.
I use the tablet with an HDMI cable connected to the projector - Chromecast is a risk I do not want when I am doing it all!
I use pads too - so do all I can to ensure I never accidentally change songs.
With keep pad on ON and keep metronome on ON (visual only)...
I start the pad using an on-screen button, observe the metronome for timing, go through the song top to bottom (maybe skip down to shorten). At the bottom move to next song (with handy auto pad change) and repeat. At end of songs I stop the pad using an on-screen button.
To split songs up into smaller sections I use '||', the opensong slide split indicator, on lyric lines positioned after the last chord. That way I display only a few lines (as people expect).
[V]
.G
Here is a first line
.G C G
Here is a lyric line ||
.G
Here is another line
.G
And another
Gives 2 Verse sections with 2 lines in each.
For those without a sound crew too - I use a TC-Helicon Play Acoustic which takes guitar, microphone and pad sound from the tablet and then allows me to process and mixing the output - a XLR line to a sound system set up (simply) for a nice room sound is then used.
Having things top to bottom using presentation order certainly helps me and may help you.
Sorry for the ramble.
welcome to the 'I do it all' mode!