The easiest way to make Long Rests matter is to tie encounters to them. For example, if you kill the goblins guarding an outpost, then go into the outpost, the goblins inside may not have heard the battle outside. However, if you long rest, that's more than enough time for the people inside to investigate and become hostile (or at least suspicious).
I would suggest a combination of the above, and restricting areas where a long rest can be initiated based on whether you can Fast Travel (because if you can, you'll just travel to an area where a long rest is allowed).
Short rests need to be more than one per long rest, but shouildn't be unlimited...there are only 24 hours in a day, and short rests are giving free healing instead of using hit dice. I suggest at least 2, and I'd suggest adding an additional short rest every 5 levels. D&D assumes higher level characters will go longer between long rests because they have more resources - a 20th level wizard has 22 spell slots, and can recover up to 7 more (4 1st, 3 2nd) with Arcane Recovery. That's before looking at items like a Staff of Power.
