Yeah, that bit was the hardest in the veteran campaign.....
I would say move from the start position asap, otherwise enemies will advance up the hill and will be a real pain - especially as they pretty much go straight for you and no-one does anything about it.
From the start, I bolted straight down the middle to the next bit of shelter (some sandbags, maybe an mg nest? It's been a while...) I think it's the same spot that Woods occupies (I tried holing up in the bunkers on the sides, but that just wasn't working for me). Once there, lay flat to heal if necessary and allow a moment for your allies to move down as well - they're not the greatest help, but they do advance with you if you wait a moment.
Then it's a question of popping up to clear enemies and running for the barrels whenever it seems safe to do so - one is very close to the shelter and I'd start with that one. Don't even think about trying to hit two barrels without shelter in-between.
Because of the constant stream of enemies, it seems counter intuitive to take your time, but so long as you don't stay at the top of the hill they don't seem to advance - allowing you to take it slow. There's also a limit to how many enemies will fill up the trenches - the one's running forwards slightly further away can be observed as scripted routes if you watch them long enough - they will only fill in the gaps in the trenches if other enemies have been taken out.
There's quite a bit of luck involved, but that worked for me - and the rest of the game is a breeze in comparison.
Good luck if you ever go back to it!