It is annoying, but stick with it - try a few different things until you get confident with it. Some of my best schemes - and I'll highlight Magestrix Rangers and Tikonov Republican Guards, are contrasting colours.
Some possible tips:
1) Go over with the white; l find it easier to tidy up with the darker paint than the lighter.
2) Use a small brush. I'm currently using 5/0 for drawing the border with the lighter colour, then tidying up with the darker using a 20/0 brush.
3) When tidying up, don't put too much paint on the brush. In fact, with the darker colour, you can almost drybrush it on - although more paint on the brush than a real drybrush.
That's how I approached my Steiner Strikers. Did the red base & grey triangle, then washed with nuln oil. Then did a couple of layers of white over the grey, not being too careful. Then redid the edges in red with a ratty old small brush, using that semi-drybrush approach. With less paint on the brush, less there to go astray. And yes, there were iterative cycles, leading to
4) Recognise when it's good enough for you. Most of the flaws you're seeing won't be visible on the table, and sometimes just positioning the mini for photos can conceal that one bit you just can't seem to get where you want it to be!
And remember - the relief on the KS minis is incredible for contrast paints, washes, etc. They are joys, aren't they?
Re stripping - all you need is a wide-mouth salsa jar (sans salsa) and lid, and a can of oven cleaner. Easy-Off is the brand I use here Down Under; just don't pick the lemon-scented, or low-irritant, version. You want the real deal.
Spray a bed of foam into the jar, drop in the mini(s), spray more foam over - make a layer cake if you're stripping a bunch! Put the lid on, find somewhere you can hold the can upside down & spray into the air until it's just blowing propellant (otherwise the nozzle can jam up with dried gunk, like any aerosol). Put the jar where it won't fume your house for a couple of hours - couple of days is no problem. I then rinse all the gunk out with water, then using an old toothbrush scrub off remaining paint gunk out of the nooks and crannies. Yes, the laundry trough is a great place to do this
