I have a sigma 120-400mm and a Sigma 150-500mm both dg Apo.
They both work well with the la ea3 adapter using almost all the same af functions as my Sony 24-105 g oss.
Af speed is very noticeably slower and they wander from time to time.
The Sony by comparison is practically instant.
When I can afford it I will upgrade to the Sony 100-400mm GM OSS lens w/1.4 tc mentioned above.
My camera is the a 7r iii, I don’t know if it is the same af system as the a7iii or not.
I also have a Tamron 1.4 tc that works with af on both but image degradation on the 150-500 is to much for me. 120-400 images are better.
120-400 refurb by sigma from Adorama was $314.00 one year ago and 150-500 was used @ $540.00 about 4 yrs ago.
Hope that helps, Good Luck!
I also have a sig 70-300 dg app uses la ea4. Prob pretty cheap now a days.

Edited about 12 times for typos.

I have the Sony FE 70-300mm f/4.5-5.6 G OSS lens and the 24-105. They compliment each other well. I am not a great fan of long lenses, but I felt I needed something longer than the 24-105. The 70-300 is very similar to the 24-105 except it doesn't have the constant f/4.0 and is a bit bigger and heavier. If you need occasional longer reach you can switch to aps-c (crop mode) which would bring it out to 450mm or use Clear Image Zoom to bring it to 600mm. You can even combine crop mode and CIZ and get useable shots at 900mm. I just tested it for you and found it snaps to focus even at 900mm in dim light.

The Sony FE 70-200mm f/4 G OSS is about the same price and is generally considered to be better IQ than the 70-300 if you can do with the shorter reach. The the $2,500 70-200 GM is out of my league.

Used Sony alpha mount versions of the 70-300 are quite resonably priced if you are prepared to use an adapter.

See the first shot I ever took with the 70-300 here. Scroll down on the post to see some crops.
https://www.uglyhedgehog.com/t-531335-1.html

I'm in the same boat as you. I have the a7 III and the 24-105, but need a longer lens for youth sports and a few other things. I pay a lot of attention to Ken Rockwell's equipment reviews and he loves the Sony FE 70-300. That would usually do it for me, but somewhere deep down inside I am bothered by the fact that my entire mirrorless arsenal would then consist of lenses that only open to f/4. I am impressed with the a7 III's low light performance, but f/4 can be limiting at times. We are both trapped by the, so far, limited lens selection for Sony full frame mirrorless cameras.

There are plenty but most are for the A mount, so you’ll need an adapter, the LEA-3 which relies on the camera auto focus is too slow and restrictive in what focus mode is supported. The LEA-4 has its own auto focus motor and is much faster but you lose the camera auto focus capability.

The Tamron 150-600 worked fine with the LAE-4 but you’re taking advantage of Aiii auto focus, using native glass is much quicker and accurate.

A warning, Sony says the LAE-3 is only for Sony A mount glass! I used it with Sony 70-300mm f/4.5-5.6 and the results were excellent, in fact I prefer the A mount glass, I had the native E mount but it was too soft over 200mm.

In sum get the FE 100-400, if too expensive, get the 70-200 f4 FE mount (a fantastic glass).
If you need longer reach get the Tamron 150-600; you can use either the A mount+LAE-4 or Canon mount+MC11.