First off, friend, you really need to know exactly how a gas-operated semi-automatic rifle works.... and why it works that way. The very best book on your rifle is the one I recommend. It is "Hatcher's Book of the Garand" (Stackpole Books, Harrisburg, PA, USA) and it was written by Major-General Julian S. Hatcher, a personal friend of John Garand AND who was intimately involved in the Garand rifle program from beginning to end. There are manuals, yes, but there is NO better book on the topic of your rifle. Another book which will really help you a lot is by the same author: "Hatcher's Notebook". Same publisher and it is back in print simply because it is so very useful and there is nothing else in the English language that covers more than a tenth of its material. This book, by itself, is an education on rifles, ammunition and the development of both in the 20th century. I first saw a copy nearly 50 years ago and I STILL find it a fountain of information. On a knowledge-per-dollar basis, it is the best investment you will ever make.
Further: Winchester sporting brass tends to be a little softer than military brass. Normally, this is actually a safety feature, but there is something wrong ENOUGH with your rifle that it is becoming a blasted nuisance.
The absolute ballistic performance of this ammunition is not far-enough out from military performance to give the problems you are encountering. Military spec for your ammo was a 152-grain bullet at 2800 ft/sec MV, pushed by about 50 grains of IMR-4895 powder, which was developed specially fore the .30-'06 cartridge. Actual weight of charge will vary slightly from Lot to Lot. This is a fairly quick powder, between 3031 and 4064 in RBR. For gas issues, you shouldn't really use anything slower than these for a Garand.
My own Garand 4337### is a peacetime Springfield in about 98% condition. It eats Winchester ammo very well indeed and rather likes it. I don't think the ammo is your problem.
Your brass looks a bit scruffy in the photo here, although I really wish it was clearer.
There were TWO models of combination tool issued with the Garand, and I don't know which model you have. One model had a replaceable chamber brush, and this is the one most commonly seen. The OTHER model had a chamber reamer which was designed specially for removing RUST from the insides of chambers. I think this is your problem. If we had the rifle HERE, we could dope this out in about 15 minutes AND repair the problem, but it isn't here, so this will have to do.
On the better of the two Garand combination tools, there is a wobbly thingie that looks like a dildo for a perverted puppy. It is split down the middle (actually, the cut is made on a milling machine with a slitting saw). It is about 2 inches long and about 3/8 of an inch in diameter. The purpose of this wobbly thingie is to ream rust scales out of your chamber. This was a BIG problem back in the days of corrosive ammunition and tropical climates with wars happening in the middle of them. This reamer is on its shaft rather wobbly in order that it will work to clean the chamber without your having to remove the barrel from your rifle.... or even disassemble the critter. It was a field-expedient fix for the problem I think you have.
What you do is head on down to the hardware store and pick up a couple of sheets of 600-grit wet-or-dry paper. Stop at the gas station on the way home and scrounge a couple of empty oil bottles from their garbage. There will be a few drops left in each and you only need a few drops for the next part of this. After that, stop at the GM dealer and get a tube of LUBRIPLATE 105. Yes, this is a type of grease BUT it is approved for the M-1 Rifle AND it is necessary in order to keep your rifle from battering itself to bits.
Cut one of your sheets of 600 wet-or-dry into strips an inch and a half wide and about 4 inches long. Wrap one of these around the chamber-cleaning reamer, folding an end over so that it anchors inside the reamer. Run this into the chamber, turning the reamer as you do so. When it's all the way in, give it a couple of turns and bring it out. If it shows any brown discolouration, you have RUST in your chamber, which is why your rounds are sticking. Put a few drops of the motor oil onto the wet-or-dry and run it back in, turn it half a dozen times and bring it out. Remove the wet-or-dry from the reamer and substitute a 4x2 cleaning patch, run that in and dry the chamber. Now run in a fresh piece of the wet-or-dry, turn a couple of turns, bring it out and inspect. If there is any evidence of remaining rust, repeat the process. When there is no more evidence of rust, you can grease the rifle up properly with the Lubriplate and take it back to the range for a test.
I think your problems will be solved.
The Garand rifle normally is the most forgiving and reliable gas-operated semi-auto rifle and it has the least number of problems.... if it is kept clean and fed right. I have NEVER had a stoppage with my rifle.... but I did luck out and get one that was absolutely brand-new. I have run into rifles with problems (as yours) but in every case the problem was down to maintenance, lack thereof, bad ammo (and this includes 99% of WW2 ammo) or just plain too much ammo down the pipe.
A point to remember if you get any surplus ammo: American .30-'06 military ammo was not all noncorrosive until late in 1953. That 1969 Lake City stuff that turned up here from 'Nam a few years ago was just about perfect, but there still is lots of old corrosive junk to be found, so be careful.
DO have fun with the toy.
Really hope this helps.