Painting over linseed finish

Surface prep is one of the most important factors of painting, it takes most of the time. If the surfaces are clean of environmental contaminates, fireplace and cigarette smoke for example or hand prints around door knobs, then you could forego wiping down the trim or places were heavy deposits of "stuff" build up.

There are "liquid sanding" compounds that dull a surface to eliminate sanding, But I can't vouch for those. Hand sanding at best for detail areas, use a pole sander for larger wall surfaces, use 120-180 grit if you plan on priming the wall again. This will be needed if you plan on changing the color to something that may not cover on one coat. Use 220 to de-gloss the surface, if the finish paint can be done in one coat. Otherwise you can see the scratch marks through the finish coat (with heavier grit), if you use gloss paint. This effort over the old paint will also remove other debris that inevitably gets in the wet paint as it is applied or while it dries, such as lint, hair and fuzz, paint runs, you name it.

Use the what may be now the not-so-reflectiveness of the once was glossy paint to determine how well the scratch pattern is on the wall.

With a pole sander, it may only take an overlapping pass in each direction (left to right, top to bottom) to accomplish this. This will help keep track of what you done already. Better than waving around all over and hoping you caught all the area.

No sanding job will be 100%, the idea is that the new paint will have enough to bite into to hold up over time.

The visual amount to look for? Picture what a comb or brush for your hair would look like if it left marks on the wall after one or two passes. Of course, what you will be doing in reality will not be uniform as my word picture suggests, but the idea is to have more scratched wall surface than glossy, with no large spots that are still glossy like it was never sanded.

The problem you are having is not uncommon. It is a basement floor so I'm guessing below grade. There should have been a layer of 10mm poly under the floor when it was poured but either way concrete leaches moisture. This is a contributor to your problem.

If you didn't thoroughly wash the floor (I mean powerwash) with a bleach based cleaner you will have a hard time getting any product to last. If you can't pressure wash, scrub as best as you can and use an etching acid or solution. Kilz will dry, unless there's moisture or dirt underneath. Will have a harder time overcoming moisture as it doesn't have near the penetrating properties as a specific product like drylok does. But for a floor, I prefer the lasting quality of a concrete epoxy.

I know it's not your most favorable thought (starting over) so ultimately it's up to you. The right prep and technique with a product specific to the job may save you the hassle of ever doing it again. Best of luck!