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I play sequences using avoids also. When it is on 1 or in other important place, I alter it (raise the fourth etc.), but in other places I leave it as it is. But it is easier without avoids.

It still sounds quite overwhelming devoting 50% of practice time to play a few phrases across the songs. For me it's qite tiring to do so for 15 minutes, not speaking about an hour or two But I admit it requires such an effort it has to be very fruitful.

I also don't catch the connection between this 'linear construction' approach and playing meaningful melodies,

I suspect this way of practice leads to such a freedom that it's much easier to play phrases from your head and not stopping because of difficulties (technical, rhythmic or harmonic ones).

There is almost no standard song without sequenced melody phrases. Sequencing is a very powerful tool in elaborating melody lines. That for it has to be practiced in any possible manner.
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Things become meaningful when you repeat them. If you are able to reapeat something transposed to another key function or vary it only in a rhythmically sense it becomes meaningful.

your delayed resolution experiment sounds great too! ... it's perfectly fine if your time is a little off. for now you have the sound of the line and how it stretched out the harmony ... next step is get the sound of those things along with the time feel you want. (and then repeat as necessary

.... for building a fluid relaxed technique, it's worth it to play the solo VERY SLOWLY and WITH UTTER EXTREME RELAXATION. Very free and out of time is fine. Literally, let gravity drop your fingers to the the notes. When you can play the notes in a very relaxed and slow way without any thought at all of playing in tempo gradually then start to play in tempo. But slowly. Don't play any faster than a tempo at which you can maintain extreme relaxation. You might find it interesting to pick a phrase or two that you like and learn to sing them too ...

Hope this all helps!

oliver nelson's improvisation book
http://ebookbrowse.com/45074734-oliver-nelson-improvisation-for-saxophone-pdf-d293950896

. . . the music of Bach, Beethoven, Bartok, Charlie Parker, John Coltrane, even some of todays electronic music is constructed much in the
same manner. The use of sequential musical devices is not by any means dry and mechanical, but can be a useful aid in the art of improvisation, with or without inspiration and immense natural talent.

keidyskos, it's worth saying more than a few times that "linear" is just a jargon word that's long been used by musical theorists. in fact, it's been used all over the place.

http://www.tate.org.uk/art/artworks/gabo-linear-construction-no-1-t00191

i'm not saying this to say the LC method isn't worth doing. If you want to do it and/or it you've been doing it and it works for you, well, then GREAT! Get going with it or keep at it! My point is these kinds of jargon names sometimes obscure essential stuff .. which you kind of point out in your post: "i ... don't catch the connection between this 'linear construction' approach and playing meaningful melodies."

Yes. Playing meaningful melodies is EXACTLY what you want!

i'm not saying at all the LC method won't lead you to meaningful melodies. What i am saying is trying to figure out HOW that method leads you to meaningful melodies or just structuring your time along the lines of that particular method may take longer than you want need to get to the "meaningful" melody part .... The thing is. You can be working with meaningful melodies right from the start!

For example. Transcription, which has been mentioned and described very well in this thread basically includes learning to hear a solo, learning to notate it, learning to play it, learning to analyse it, using it to inspire your creativity. And as you do all that you're DIRECTLY IN CONTACT WITH MEANINGFUL MELODIES!

Yet, Fred Hersch, who many consider to be at the apex in the jazz piano food chain (if you haven't heard FH, at least check out his recordings ... the last two are from live sets the Village Vanguard). He's said many times (interviews on the web) that he found no value in transcribing. So there you go. A major artist who didn't transcribe to learn jazz and has said he doesn't see the point in it. FH recommends instead very, very, VERY serious directed listening to all the different versions of jazz repertoire that great artists have produced. That is pretty serious!

Or, instead of transcribing, sing great solos (sing along with the recording). Sing, as in: FOCUS ON THE SOLO AND SING IT LIKE PLANET EARTH DEPENDS ON THAT SKILL!

Or get out whatever classical scores you have. Mozart, Bach, Chopin, etc. They all wrote INCREDIBLY MEANINGFUL MELODIES! In most cases, with just a little of bit of creative phrasing, those melodies will work directly in jazz.

A great book about listening to and creating music is "The Listening Book: Discovering Your Own Music" by W.A. Mathieu (he has major credits including writing for Stan Kenton). The chapter in the book: "What should I practice?" He says: "It doesn't matter what you practice as long as your energy for it is HOT [... my emphasis ...]. It is the quality of your practice that matters ... "

Mathieu isn't saying just be unstructured and do whatever you want ... you gotta read the book to get that larger context and to see how he says it. But the point is this: If you have limited time (and most of us do), one good way to go is practice whatever you can put your whole 100% focus on. Get your mind, ear, and body RESONATE to when you practice. If your interest is in playing and creating meaningful melodies, then focus your practice right onto that. Find examples, in recordings, scores, transcriptions Work directly with them! Play them, sing them listen to them, copy them, transpose them, think about them, discuss them, quote them, Play them in all keys. Play the major ones in minor keys. Play the minor ones in major keys! Write counterpoint against them. Re-harmonize them. Take liberties with them. Learn them so they just about feel like you own them!

Work with meaningful melodies -- you'll soon be making them yourself.

Just my opinion ....

Hope this helps!