Gaming is a hobby and much like other hobbies, not every dollar spent on gaming is equal. I've spent money on games before that I've never played, I've sold games at a loss that I've never played, and I have a collection of sealed games waiting for me to break them open. Digital games aren't the issue, but they can highlight the issue since you don't have a physical game on the shelf that you can say you bought for collecting purposes even if you never play it.
That said I think creating a budget is key towards getting rid of anxiety. I try to limit my spending to a set amount each month that I'm comfortable with. Then, next, is to focus on playing games over looking on websites or looking for deals for games. I find I'm much happier when I'm actually playing through games instead of accumulating them. Usually I limit myself to like one or two days a week to check deals on games and post on websites such as this.
I only buy digital games when I want them. I usually wait for sales to buy them. Ducktales on the Wii U for $7.49 was something I was waiting on and I didn't hesitate to pick it up when it hit that price. Yes, there are concerns that paying for something that I'll either get minimal use out of or may not get to. But the reality is that digital content will still be playable for a long time even if it's technically not owned. I have some digital content on my PS3 that I need to get through, but it didn't stop working when the PS4 came out and it won't stop working when inevitably PSN stops working on the PS3.
I don't have any rules about trying to get through my backlog. If I'm playing 5-10 hrs a week of games I'm generally happy with my time spent on gaming (family time gets priority and makes it hard for me to get more gaming than that). If I want to play an oldie but goodie game and leave my backlog untouched, that's fine with me. Sometimes I'll crack open a new game and get enamored with it and make a lot of progress on it. Sometimes I won't be that excited about it and I'll either put it back in the backlog or sell the game. I don't want to force myself to play through games that I'm just not interested in.
I know that sounds counter-intuitive, why buy something your not interested in playing. But I buy games that sometimes I don't enjoy. Sometimes I may not be in the mood for a certain type of game so I generally try to re-backlog games I didn't like to make sure I try them again or a couple more times to make sure I didn't like them. I sold alot of games during the Gamecube era that I wished I hadn't and I ended up buying them back. So I'm not in a rush to decrease my backlog for the sake of decreasing it.
But really I think you just have to get used to the idea that hobbies cost money and it's not always money well spent. But sometimes you may buy a game for $5 that you get an incredible amount of playtime out of and it's money incredibly well spent. It balances out. Just have a budget of what you are comfortable with spending on games and stick to and be comfortable with that. Most hobbies are expensive. As someone who has hobbies like hockey, snowmobiling, motorcycling, I know that hobbies can be very expensive. You just need to get comfortable with your overall enjoyment you get from the hobby and your overall spend on it.