I’ve just done a little listening and searching.
In the song I heard 3rds, 6ths and triads for the solo.
I have watched a few lessons which confirm
Have a visit to and a read of - with a view to doing some guitar work, not just reading - this topic which happens to cover what you need also: 3rds in the key of D. 3rds & Thirds - What Are They? Playing & Having Fun
As to the general approach with two guitars and you taking a simple, melodic approach to soling that still keeps the sounds of the chord movement going …
Yes, yes, yes.
The triads give you chord tones and if you target the notes within, but as you say played not as chords but as notes, with all the various ways that you can attack and approach and give dynamics to single notes, then you are soling over the changes and keeping the changes clear.
If you want to improvise a solo and make up your own thing for Maragaritaville, try this.
Are you familiar with a major pentatonic extended pattern that stretched across multiple frets? If you are great. If not, have a little try out and see if it falls nicely under your fingers.
In general, strings with two notes will be fingers 1 & 3, strings with three notes will be fingers 1 and 3 with a slide up using finger 3. But you may find it works better using different fingering.
Mix it up with the triad chord tones that you have been exploring.
Here are some diagrams to help.
I hope that helps.
Cheers
Richard 