It's really hard to give you any relevant advice without knowing your approximate skill level. Still, let me provide you with some general pointers:
PROBLEMS WITH CVs:
1. The CVs need to find you first. In this case two things work tremendously in your favor and one against you:
- planes are extremely visible with the exception of IJN torpedo bombers (that can't hurt you if you maneuver right). You're going to see them coming from far away
- playing a DD against planes is like having an invisibility cheat - they need to be almost right above you to spot you. This is a really stupid idea of WG (they did that instead of removing the third factor that works against DDs) but it makes both finding a DD really hard, as long as the AA is off
- (this hurts and this is what should've been addressed instead of giving DDs ridiculously low air detection) planes know that they are spotted so a CV player with half a brain can easily realize that SOMETHING (presumably a DD) is in the area - this makes it difficult to relly operate solo because the CV doesn't have to spot you to realize you're around... which may prompt a scouting sweep of the area if there are no other ships
2. You don't really need to be behind friendly ships or right next to them to benefit from their AA. You can be ahead and just retreat towards them as need arises. How far ahead you need to be depends on your ship, the friendly ships and overall situation. Also remember that after attacking you, the planes often end up on the other side of you... and if they attack from the front and there are a couple allies behind you? Well, there won't be a second strike from THAT squadron...
3. CVs can reach you anywhere on the map but they don't teleport. Even the fastest planes take quite some time to cross the map - if you see the enemy CV working on the other end of the map, then he won't get to you instantaneously. He'll need to either fly to you the planes you saw or go back and start a new squadron. What's more, CVs don't really like too fly huge distances - a sensible CV stays as close to the allies as he believes he can get away with AND focuses his attention mostly on the same flank. This happens because time spent flying to target is time spent doing nothing productive. CVs avoid flying their planes to the opposite edge of the map unless they have a really good reason to do so (note that someone attempting a cap with no support might be seen as just such a reason)
4. CVs don't spend their day looking for DDs. They go searching for you when they have some idea as to where to find you AND some hope that this won't be in close proximity to some heavy AA platform. This includes the opening phase of the game - DDs going straight for the caps, far ahead of teammates that could provide AA cover, are both vulnerable and predictable. Don't do that. If you want to push a cap early, make sure you don't get too far ahead and can retreat towards them. Usually, however, it's just not worth it to attempt an early cap like this. Know where the Radars are, where the planes are active, that friendlies aren't too far - and THEN cap. DDs are still the best cappers, BY FAR, but they do so by pushing in just that bit closer that would be a fatal overextension for everyone else - but isn't for them. Ignoring your allied fleet and going to hunt/cap all on your own has a good chance of backfiring on you quickly in a match with CVs
5. Learn to manage your AA. Even DDs (at least the high tier ones) don't need THAT much time to dispatch fighters and setting up your priority sectors right makes it much faster still. And then there are the DDs that actually enjoy good AA and can properly hurt the attacking squadrons as well. Again, priority secotrs are important for that. Plus, of course, there's the matter of switching AA off completely. I mentioned it when I talked about getting found - but it actually serves a purpose as defensive measure as well. Many squadrons (most notably rocket planes, the main anti-DD weapon of IJN and RN CVs) to deliver a good strike (or any strike at all) need to start their attack run from beyond your basic air concealment range. Which means that when they set up the attack, they might not see you at all (unless they are spotting you with fighters or somebody else spots you for them). If you manage your AA right and don't sail too predictably, you can make it much harder to land rockets on you. Ito doesn't work all that well against HE bombers, though - the USN ones don't need to start the attack run too far away and the RN ones, while not nearly as effective against DDs, allow quite a bit of adjustment during the attack run, making it less crucial to know the precise position of the target at the moment of starting the attack run.
PROBLEMS WITH RADARS:
Well, the reality is: you need to locate them and avoid them. Also, especially when you're not sure of their position, you should really avoid ever stopping in smoke. With Radars nearby, smoke doesn't conceal your position as much as reveals it - guess what a Radar cruiser is going to do when they notice a smoke cloud appearing within their Radar range... And, of course, when there's a cap covered by enemy Radar, you should probably give up on taking it at that time. There are some more advanced tricks where you can try and bait Radar by pretending that you want to cap - but they aren't that easy, you need to be pretty good at assessing the danger and judging by your post alone (since I can't see your stats) I doubt you are up to things like this for now.
As a final note: DDs aren't an easy class to play. They are faster than BBs and stealthier as well as harder to one-shot than cruisers but to be effective, they also need to get closer - and that amplifies the risks involved. And yes, on high tiers the abundance of Radars along with the fact that CVs are only endangered rather than extinct right now means that DDs are in an even trickier position. They can still be rewarding and the relatively low numbers of DDs actually serve the class well (every DD kill counts more for a DD-hunter DD, there are less DDs to prematurely spot torps) but striking the balance between the appropriate level of aggressiveness (to be a threat) and caution (to not die early in a dumb way) is harder than ever. Still, if you can do that, you'll see yourself having a lot of influence on the outcome of the battle.