It's a pretty common tendency to fix problems by buying more eyepieces, but that's rarely how it works in practice. There are a few steps you need to take before buying more glass.
Check collimation first. This should be your first item on the list, full stop. The performance of the scope decreases rapidly when it's out of collimation (for solar system objects). Check the documentation that came with your scope. Also check these links:
If the scope is not in good collimation, a new eyepiece is a waste of money. All the eyepiece does is magnify the real image generated by the scope in the primary focal plane - so if that image is out of whack, then no eyepiece will give you the performance you're looking for.
Check collimation every time you use the scope. With practice, it will end up taking only a couple minutes.
It varies with time and location. Sometimes it's better, other times it's bad. Go on the CDS site, search for your location and use the Seeing forecast; good seeing is represented by dark blue, bad seeing by light blue or white.
Plan your observations according to the forecast.
Seeing only affects fine-detail objects like the Moon or the planets. It does not affect faint DSOs (galaxies, nebulae, etc) because your eye does not resolve fine detail on DSOs anyway.
If the primary mirror is not at the same temperature as the air, "local seeing" conditions (vortices of air on the primary) will blur the image. Take the scope outside 1 hour before you start observing and let it acclimate.
For a scope 8" or bigger it's often a good idea to install a fan on the back of the primary to accelerate the process - but get everything else done first.
Only affects DSOs (galaxies, nebulae, clusters). It does not affect solar system objects such as planets. The only remedy is to load the scope up in a car and drive away from the city.
However, all Messier objects are visible in most amateur telescopes even from very light-polluted places like major cities. They just don't look as spectacular as from a dark sky place.
Make sure the scope is actually focused properly. This might be more tricky than it seems. Use a star nearby and adjust until the star is as sharp as possible.
You could also make or buy a Bahtinov mask, which will give you proof positive of perfect focus (adjust focus with the mask on, until all 3 lines intersect each other in the center).
With all of the above taken care of you should definitely see Jupiter's equatorial belts. They become visible even in 2" of aperture, and a slightly larger instrument should show them pretty clearly, even in a modest eyepiece. Under good seeing, with a perfectly collimated instrument, you can even start noticing the curls and the vortices in the belts even with an aperture as low as 4" ... 6".
Jupiter getting more and more blurry as you increase magnification is normal. It's not a sign that the eyepiece is bad. What's not normal is not seeing the belts in a 10" scope. That indicates that either collimation is bad, or seeing is exceptionally bad, or both; typically it's collimation.
Your eyepieces should actually give decent performance in the center of the field of view.
You also need to let your brain-eye system adapt to the image. You don't have to observe from a very dark spot, Jupiter is bright enough to not require perfect dark adaptation. But your brain and your eye need a bit of time until they can extract all details from the image. Let it drift through the field, stay relaxed, follow the image and just take it in slowly. You will start seeing details after 3 or 4 minutes that were not visible at first (assuming good collimation, seeing, etc).
Now on to the eyepiece. Your instrument is a f/4.7 newtonian, 1200 mm focal length. A 10 mm eyepiece will give you 120x magnification. This should not be too much even under bad seeing - provided the scope is collimated.
I've observed Jupiter a lot at 1200 mm f.l. in a smaller instrument. I've found that a 8.8 mm eyepiece (136x) was a reliable workhorse, never needed anything with less magnification. When seeing got better, I could push it up to 6.7 mm (180x) or even 4.7mm (255x). But speaking in general, Jupiter is low contrast and does not benefit from pushing the magnification too hard.
Mars OTOH does very well under very high magnification. But that assumes your collimation is top-notch and seeing is very good.
Provided that you take care of collimation, etc first, you could try and get better eyepieces. The ones that came with your scope are not that bad actually, they do fine in the center, it's the edge performance that's weaker. But if you do get new eyepieces, you might as well get something better. I've found that the Explore Scientific 82 degree series provides a good bang for the buck - not too far from the performance of top glass, but much cheaper than top prices.
But the options here are very numerous, so do your own research.
You might also benefit from having a barlow, maybe a 2x or a 2.5x. However, the much higher magnification generated with the barlow will not work unless collimation, seeing, etc are all taken care of!
It's a little difficult when you juggle collimation and seeing at the same time, one you can control (collimation), the other you can't (seeing), but realize that this will take some time to figure out, so go one step at a time.
Also understand that this is not the kind of problem that is solved by simply throwing money at it.