Hi!
They begin playing extremely defensive themselves. Examples of this behavior includes running away and evading my every move or camping at the other side of the map. I've had several fights where they seriously just dodge and jump/run fast away from every attack I make. I simply cannot keep up with the speed as Zelda is very slow.
This sounds very much like you're trying to chase them. Never chase these characters because they are looking to make you overextend and whiff punish you, something that is very easy for Zelda to do. As you've found out, she is too slow and her burst options aren't safe. I find Zelda's speed sits in a particular niche where you are tempted to do things you would with fast characters, but she should probably be considered more along the lines of a traditional heavy character.
What you need to do in these situations is severely limit their movement options and slow them down to Zelda's pace. The easiest and best way to achieve this is through Phantom Knight. I know people tend to write it off as a gimmick and say good opponents are too smart for that, but it is very powerful with the proper spacing.
Consider that the final charge's hitbox swings up roughly 4 big squares vertically in the Training Mode stage. That's enough to catch every full hop I can think of right now. The hitbox is a bit janky in that it doesn't seem to be active throughout the full back swing, but if you space at five big squares horizontally from your opponent, you'll be able to catch a neutral short hop or full hop. On Battlefield that would be roughly in the center of the stage.
That means that the opponent would either have to commit to a defensive option like shield or roll or they would have to jump forward or backward or burn their double jump. If they are ledge camping they can't jump backwards. The final charge detethering from Zelda means she is free to cover every other option, such as roll or shield.
Consider Fsmash's range is roughly 2 big squares horizontally. If you walk a couple steps after fully charging Phantom then you can catch everything from roll to a forward jump or directional airdodge. If your opponent tries to go over you entirely, Up-Tilt, Up-Smash, and Up-air are all available to catch airdodges, double jumps, and aerials. Should he try to directional airdodge down through Phantom, dash attack, grab, and running Up-Smash is available with the right timing.
Learning how to zone with Phantom is tricky and probably the hardest thing about her, but it is essential. Maybe not for all or even most MUs, but these ones for sure. It's an extension of her neutral, so the ability to read and react to your opponent is essential. You need to understand the options your opponent has at each distance and out of each counter option they do against Phantom. That ups the learning curve quite a bit, but your reward is arguably the best projectile in the game that borders on overpowered.
Learn optimal Phantom positions to be in on every stage and focus on taking those positions. Center stage is powerful on Battlefield and will cover almost every option your opponent takes.
As far as the general idea of dealing with defensive play goes, you need to be even more patient than your opponent. I don't know what ruleset you regularly play on that leads to frequent near timeouts, but if they are that frequent, you are either not playing a competitive ruleset (3 stocks, 7 minutes) or you're not optimizing your advantage enough. Or you're playing online, which is another beast entirely. Timeouts are fine so long as you have the lead, so I wouldn't worry too much about things going near the limit.
Why I advocate for Phantom is that it allows you to safely poke outside of your opponent's whiff punish range. Being able to safely fish for a setup is huge on camping opponents, because it is only a matter of time until you get lucky or they screw up. That signals to them that they probably shouldn't let you freely spam it for long.
But in the case that you can't use for Phantom for whatever reason, you need to think smaller on this. Don't think of things in broad terms like he airdodges so you have to chase and punish or be faster than him or whatever. Instead, think of smaller victories like gaining more space and stage control or being able to safely encroach into his space and be in position to punish his mistakes. Those little victories are ultimately what setup big plays.
If you are fighting for center stage and he jumps over you, don't chase him and try to punish. Turn to face him instead and take advantage of the fact that he just gave you center stage, where you now have the ability to Phantom zone. I see a lot of Zelda players blindly chase their opponent in this situation and not only give up the advantage they just got, but get whiffed punished as the icing on the cake.
If your opponent is camping on a bottom platform and is repeatedly jumping, don't chase with an aerial and don't try to pressure him with Up-Tilt (-15 on shield means he can escape easily). Move as close to the platform while still being roughly at center stage instead and watch him closely. If he's alternating between short hop and double jump, watch for patterns and pounce when he's going to use his double jump. Zelda's Up-air is big enough she can get away with only a short hop or full hop Up-air in most cases. He can't really do much else safely there and has to chose between dropping through the platform right in front of you or going on the attack.
I know that it is difficult to see this with the illusion of freedom Smash gives, but it's not that much more free than a traditional fighting game. What might seem like complete movement freedom for these quick characters is really just them giving up space. You need to be the choke point and cut off that movement after they've given up space. Be it with Phantom or with Zelda herself.
And that's what you ultimately need to do. You need to tyrannize your opponent with space. Even if you aren't actually doing anything, just making them think you could pounce at any moment is already putting pressure on them. Find where you can halt them for maximum stage control and keep them there. Think of a large bubble around the stage with the focal point being center stage. If you can put a wall around there, your opponent will always in some form of space deficit. And the more he pulls back, the more you can push that wall towards him and the less space he has to work with.
What you are describing sounds like kiting, which is something that isn't easy to deal with for even other fast characters. It's not a Zelda exclusive issue. And the way to break kiting is what I've laid out above. You just need to be patient and aware of the overall state of battle.
