This is a long post, but the feedback I've received from referees watching my games is that I am pretty good with judging offside without an AR.
I don't necessarily find that getting wider helps. It kind of depends on how they're playing. The worst is when one team constantly has the striker pushing the offside line on the opposite side of the field to the sweeper, when the opposing team's striker is basically being marked by the sweeper all game long. One team you can spot the offsides, the other you can't. Really frustrating.
Try getting wider, see if that helps. It may well depend on how the attackers are positioned against the attackers. When you're by yourself in particular you really have to be prepared to drastically change your positioning to suit the needs of the game. I've had games where I've basically hung out with one team's sweeper for almost the entire match (or practically adopted an AR's position) because that was what I needed to call all game and absolutely nothing else was happening. For that game, and that game only, it was great positioning. Any other game with that positioning and you'd think somebody had snuck something into my gatorade. Even games where I'm not spending a lot of time doing that, sometimes I will be roughly in line with the 2nd last line of defence, but of course that's judged against what I'm putting myself out of position for. However, that position isn't necessarily great because I can't constantly watch that 2nd last line as closely as an AR could in many cases (sometimes I can), and if I have to look left and right because there's attackers on either side, then it doesn't work at all.
What I've found works is, if I'm looking up at the offside line, I draw my eyes in a line along the grass, left to right. Almost envisioning a physical line, but drawing my eyes directly across the field. Doing that you have a pretty good chance of keeping it in a straight line, perpendicular to the touch line. And I find my mind sort of seems to register that line, so even for a few moments it makes it clear if the players are moving. It's sort of hard to explain, but it helps. And of course, if you can preempt play then you can start to do this in advance, and position yourself in preparation as well. Sometimes I'll look up and realise that I can't tell who's on and who's off, so I need to run somewhere else - and fast.
Offside is physically impossible to get 100% right even with 2 excellent and experienced AR's. By yourself you have no chance. You'll call onside attackers off, and vice versa. Games will be decided on your incorrect offside decisions, guaranteed. All you can do is be the best you can be.
And of course, a player can by 15 yards on and the defending team will all scream bloody murder that he was off, and vice versa. Players are weird. Naturally the closer you are to the offside line the easier the decision is to sell, though I don't think proximity has as much of an impact here than in other decisions.
Where I originally refereed we didn't use CAR's at all. Where I most recently refereed we did use CAR's, and they did offside. Naturally I'd try to get a measure early in the game for how good they were at that part of the role and adjust my positioning to suit - knowing that I probably want to be in a 'more perfect' position, or one where I can 'sell' better, if I want to overrule a CAR than if I was by myself.
And lastly? Have the balls to back yourself. Believe it's on when all the defenders are standing still waiting for your decision? Stick with that. Believe it's off when the defenders haven't even thought to appeal? Go with it. Not sure enough to make the decision? So be it.
