The AF sensor(s) in most DSLRs is located in the "floor" of the mirror box. The mirror you see when you have the lens off is actually semi-transparent and there's a second mirror behind it that redirects the image to the "phase detection" sensors that are located down below. I believe your 650D has 9 AF points (sensors) and that all of them are the higher performance "dual axis/cross type". Some cameras only have one or two or three of that type and the rest are lower performance "single axis" sensors (some oriented vertically, some oriented horizontally).
More often than not, problems where AF performance has been okay, but drops off with multiple lenses, it's a problem with the electronic contacts needing cleaning. Did you use isopropyl alcohol (also called "rubbing alcohol") to lightly dampen a rag and wipe the contacts on the rear of the lens? Finger oils and similar on them can interrupt the very low voltages used to communicate between camera and lens.
If it were just one lens fouling up, I'd suspect that it's AF system had some fault. That's DOES happen. But if you are seeing it happening with all your lenses, it's unlikely lens-related, more likely camera-related. It might be as simple as a large chunk of dust has settled over the AF sensors and is partly obstructing them. If you have a "bulb blower" to gently puff away dust from the camera, remove the lens and try that... But DON'T go poking around inside there. It's all too easy to damage the mirror or the focus screen or the shutter. If a gentle puff of air doesn't clear things out, leave it to a professional to check and clean the camera, then test the AF system.
Another all-too common problem is incorrect camera setup.
First, your Canon has three different autofocus modes to select between: One Shot, AI Servo and AI Focus. One shot is for stationary subjects. It acquires focus, stops and locks, and gives you Focus Confirmation. AI Servo is for moving subjects. It continuously updates focus and never stops (until you release the button). Since it never stops, there's no Focus Confirmation with AI Servo. Finally, AI Focus isn't really a focus mode at all. It's automation. The camera is supposed to decide for you whether or not the subject is moving, then switch to use either One Shot or AI Servo, whichever is appropriate. This seems to cause a slight delay and doesn't always choose correctly or may not switch properly if a subject starts or stops moving. I avoid AI Focus entirely. It also might be a clue that the most pro-oriented Canon cameras don't even have AI Focus choice... they only have One Shot and AI Servo. Note: If you select a scene mode such as "Sports" ("running man" icon), those are "super automated" and will likely override any AF settings and may force you to use AI Focus.
Next, you can choose between All Points/Auto Selection and leave it up to the camera or manually select a Single AF point. The latter is more work for you, to keep it right on the subject, where you want the lens to focus, but is by far the most consistent and accurate method of autofocus. I use Single Point the majority of the time, only make limited use of several different multi-point patterns my cameras offer, in very specific situations where they can work well. (Your camera only has two choices: All Points or Single Point.)
A "do it all" lens such as that 18-300mm typically doesn't have a very high performance AF drive system in the lens itself, either. I looked it up and it appears to be one of Sigma's "micro motor" lenses, which are slower and might hunt more than their higher end, "HSM" or "hypersonic" focus motor lenses (similar to Canon's USM or "ultrasonic").
Ambient light conditions, oblique light striking the front element of the lens (use a lens hood), low contrast or lack of much detail on the subject itself, all can make AF struggle to "lock on" and track subjects, too. That's particularly true with a variable aperture zoom that drops to a relatively small, dim f/6.3 at the longer end of it's range. It can be further exacerbated by adding a filter to a lens, especially one that's fairly dark and further reduces how much light is available for the AF sensors to work with (such as a Circular Polarizer, which reduces light by up to about 2 stops). You didn't mention what other lenses you have, so it's hard to say what may or may not be expected from them. Some of this is simply out of your control, of course. But other things you can work with and change, hopefully improving AF performance.