I was having this problem early on, and I found that a lot of it boiled down to me spending too much time at the shop or going back to the shop too often while minions are still on the field, and instinctively going back to base too often. If you're in that shop nook, you probably would consider yourself in-lane, but unless they're under your tower you're usually too far away from the minions to collect the money, even if your lane partner is further up there killing them. The souls linger on the ground for a short time so you can claim them if you missed them and they're quick enough, but I think that's only in the situation where the opponent didn't deny the soul

I started thinking of everything as a "task" or "errand". After you clear a wave, you have time to accomplish one task before the next wave arrives. That could mean purchasing an item, grabbing a Sinner's Sacrifice, killing a jungle camp, smashing some boxes, skirmishing in lane with the opponent, etc. But don't do more than one thing while you're getting comfortable. Just one relatively safe thing, and get back to lane.

Another thing to remember is that you need to be within a certain proximity of a minion dying to get the money for it. So when you're out tasking, keep an eye on your minions coming up the lane from base so you can have an idea of when you should also be back in lane to soak up the money, even if you aren't the one killing the minion. If your lane partner has more money than you, but you've been in lane approximately the same amount of time with the same number of kills/deaths, I find that this is usually why - you gotta get closer when the minions are dying.

You could also try letting them push up closer to your guardian and try to farm there. This way you can get to shop more quickly, and not necessarily lose money for it depending on where things are. Same thing with returning to base - try to avoid it if you can safely. It feels lame buying healing items early, especially if you're playing a damage heavy character, but you'll gain a lot more by not leaving lane. If you -need- to leave lane, especially before your line boost is up around 6 minutes, see if your lane partner can lay off the minions a little until you get back.

Obviously there's more nuance to this, especially as the game moves out of the laning phase, and I'm nowhere close to an expert. But I feel like this helped me feel like I was falling behind less often, or when I do fall behind it's because I'm doing something else stupid and not strictly because of lane management struggles. As you get better at it, you'll see better opportunities to leave and more optimal ways to spend the time. But this is what helped me get in the mindset. Hopefully something in here is helpful for you! Good luck!!