I think the correct approach to this problem is to make new liners. Possibly only two of the four pieces need to be re-done to get everything tight enough*.
Now hiding mistakes is a standard part of woodworking :-) but so is remaking components that haven't come out quite right.
And as these are such simple pieces this shouldn't feel like too much work. If the thought of the wasted material is part of the resistance to the idea the existing ones don't have to be discarded — you could deliberately size your next box so these liners will be slightly oversize, so you can shoot them down to a tight fit.
If you do want to try shimming as per the Comments I'll suggest a way that you might not have thought of. Use stout paper or thin card stock, cutting rectangles the full length and depth of the inside of the walnut box but narrower so that they lie below its lip.
A paper/card inner lining will effectively thicken the walls of the box, pushing the wooden liner material inwards in both directions and should easily take up the slack enough that the liners are good and tight like they should be. They'll also provide a shadow line between box and liner which you may find attractive.
The thickness of each piece of paper/card needs to be much less than 1/64" (0.4mm) to take up slack of a thin 32nd.
Unfortunately this will probably cause the lid to be a slightly looser fit.
For next time
Make the box first so that you're taking measurements from the real thing, which may not be perfectly square. It's best not to go from the ideal dimensions given in a plan, or from calculations in your head based on the outer dimensions of the box plus twice the stock thickness.
Then cut your liner pieces slightly over-length and shoot one at a time to a tight fit in the box by planing in a suitable shooting board. If you didn't size the current liner pieces by shooting this may account for any difficulty in getting the liners just right, methods using power tools are fraught with the risk of trimming just a smidge too much off.
But regardless of the trimming method you use sneak up on a good fit — trim a bit, test fit, trim a bit, test fit etc. Remember we have multiple opportunities to shorten pieces that are too long but no opportunity to lengthen them once they're too short!
Just to note, longer liner pieces in particular may need to be flexed ever so slightly to get them in so they're a super-tight fit.