Well, besides a better tripod, there may be some other things you can do:
1. Don't "hit" the shutter button. Don't "jab" or "stab" at it. Instead "press" or "squeeze" it.
2. If shooting stationary subjects (such as a scenic view), use the camera's built in self timer to release the shutter while not touching the camera at all.
3. Increase your shutter speed. This will require boosting ISO or using a larger aperture or a bit of both, but faster shutter speeds will help avoid camera shake blur.
4. Use a remote release such as the Sony RM-VPR1 to trip the shutter while not touching the camera.
All that said, a more stable tripod may still be needed. You generally "get what you pay for" with tripods! Cheap ones don't work well or hold up for very long. So you end buying another and may even spend more in the long run! A poor quality tripod also may not be much fun to use, so you'll be less inclined to do so and more likely to just leave the tripod at home.
Depending upon which version you have, that camera weighs under two lb. to about 2.5 lb., so you don't need a really heavy duty tripod. Still, you need one that's stable, don't want something too light and flimsy. A good tripod can last a lifetime and be used with future cameras (one of my tripods is about 40 years old and still works fine, another I've used for around 20 years). While you don't want to "break the bank", it's possible you should spend more than you think, or you won't improve your situation.
The most stable tripods use the least number of leg sections and are tall enough to use comfortably without having to raise the center column at all. I also prefer the "twist" type leg locks, rather than levers or thumbscrews which both protrude and can catch on things. The twist type also hold up well and are pretty maintenance free. The lever type get loose over time and may need adjustment or replacement. Carbon fiber tripods are lighter than aluminum and CF also can help by absorbing some fine vibrations that metal will transmit, however CF costs more.
Let me give you some examples:
Benro FTA28AB1 Travel Angel Series 2
Aluminum Tripod with B1 Ball Head currently sells for $250. It has 4-secton legs, reached 53" tall without raising the center column, weighs 4.6 lb. and can support 22 lb.
Benro FTA28CV1 Travel Angel Series 2
Carbon Fiber Tripod with V1E Ball Head is selling for just over $390. It's very similar to the above, except it's about 3" taller and about a half lb. lighter.
There are also lighter duty versions:
Benro FTA18AB0 Travel Angel Series 1
Aluminum Tripod with B0 Ball Head currently costs $199. Design is similar to above, except it's not as tall at 49" without raising the center column and rated to support about 17 lb. and weighs 3.9 lb.
Benro FTA18CV0 Travel Angel Series 1
Carbon Fiber Tripod with V0E Ball Head costs $297. Again, design is similar to above, except for 51" height w/o center column, 17 lb. capacity and 3.4 lb. weight.
All four of these tripods come with a ballhead that includes a standard Arca-Swiss style quick release (the most universal type). They also also all are designed so that the legs "reverse" fold for as compact storage as possible (this limits them to using ballheads of certain sizes, but correct ones are supplied with each). They also all come with a carry bag or cover w/strap.
https://www.bhphotovideo.com/c/compare/Benro_FTA28AB1_Travel_Angel_Series_2_Aluminum_Tripod_with_B1_Ball_Head_vs_Benro_FTA28CV1_Travel_Angel_Series_2_Carbon_Fiber_Tripod_with_V1E_Ball_Head_vs_Benro_FTA18AB0_Travel_Angel_Series_1_Aluminum_Tripod_with_B0_Ball_Head_vs_Benro_FTA18CV0_Travel_Angel_Series_1_Carbon_Fiber_Tripod_with_V0E_Ball_Head/BHitems/1157840-REG_1157837-REG_1157841-REG_1157839-REGAll four of these also have 4-section legs. I prefer 3-section... better stability, larger diameter lower leg sections, faster to set up... but also not as compact when close up. I do have one tripod with 4-section and it's fine (but it's a heavy duty, extra tall model). I'd avoid tripods with 5, 6 or more leg sections. Those will be super compact... but really spindly! And they'd take about a week to set up
You can find similar tripods from other manufacturers or with different features if you prefer. The above caught my eye and seem pretty good values for tripods that look like they should work well and last for a long, long time.
Finally, be careful about the quick release systems some other manufacturers use.... which are proprietary, incompatible with any other brands, don't work as well as Arca-Swiss and my be difficult to get add'l parts for. Maybe you don't want a quick release system at all... but they are nice (even necessary for some things). I'm pretty sure the above tripods come with one "universal" mounting plate. For more secure mounting and convenience, you might instead want to buy an L-bracket to use on the camera.... which allows it to easily be swapped back and forth between horizontal/landscape and vertical/portrait orientation, while keeping the camera better centered atop the ballhead. I see a brand called SmallRig makes a pretty neat bracket for some of the RX-10 models, which costs $79 (it includes an off camera flash mounting platform).
https://www.smallrig.com/smallrig-l-bracket-for-sony-rx10-iii-iv-2230.html Hope this helps!