This really isn't as hard as you might imagine... don't get me wrong, the enemies are harder and ammo/vigor's etc. are fewer, but I never ran out of anything. The key is to take your time and search everywhere. Some other choices I made that really helped me are:
Upgrade the Possession Vigor as soon as you can - this was my life saver... I would often hide from a large battle and just hit possession on anyone who got near me. This left only a few enemies to finish off.
Sniper Rifle and Shotgun or Handgun should be your weapons of choice - of course, this is down to preference, but I found the sniper rifle was great for those enemies that wouldn't come to me, and the shotgun or handgun was great to finish those that did! I only upgraded these three guns during this playthrough, as money is quite scarce.
Return to Sender Vigor is vital for Lady Comstock and the last battle - again, another preference that some might not like, but I found that this was amazing for the two hardest parts in the game. I'd save your cash and upgrade this vigor fully if you can - use it whenever your shield is depleted on Lady Comstock and lay traps in front of the generator that needs protecting in the final battle.
Upgrade Shield and Salts first and Health Last - I found salts invaluable for the Possession vigor and Shield for sustained protection... if your health is knocked down, there are usually pick-ups (such as food) that are close to a battle and Elizebeth will often throw you health packs or can open a tear nearby which you can use.
Winter Shield clothing is great for when there are skylines - or so I've read, unfortunately, as clothing is random, I didn't get to grab this piece (and I thought I had picked up all clothing upgrades!). If you do get this, then be sure to equip it for the last battle.
I hope this helps.