In short, I either put the seafood in at the beginning and it gets cooked to rubbery perfection, or I add them later and they don’t get flavored with the rice.
What am I missing?
How are you trying to cook them? Are you putting rice and the seafood in a rice cooker? Can you cook your rice with seafood stock instead of water? What kind of flavor are you adding to the rice that you want the seafood to have?
Need more details please. 
Add a bit of fish stock (you can buy it frozen) to the rice while it’s cooking. Add the seafood at the end and cook briefly. You can make a fast fish stock with shrimp shells or pieces of fish. Just use a stick mixer to thrash it up good, bring to a boil and strain off the shells/bones.
Simulpost!
Why would you want your seafood to taste of rice?
Cook the rice, then the seafood.
If you want your rice to taste of seafood, cook it in fish stock.
Shortcut: Throw the shrimp shells in while the rice is cooking. (Or a chunk of fish skin, plus the spine if you have it.) Then discard the flavoring fragment, and add the quickly-cooked shrimp (or whatever) to the rice at the end.
Actually, I have a good source of fresh, free fish heads. I often make my own fish stock that I use for many things.
Where I am failing is making a non-paella where I have different kinds of seafood (fish, shrimp, octopus being the most common) plus some veggies and the required spices.
I want to seafood to taste of all the spices and veggies, and the rice to taste of seafood.
If I add the seafood later, it gets cooked well, but it doesn’t catch all the flavor from the vegs and spices, and the rice tastes of veggies but not of seafood. If I add them early, everything tastes the way I want it, but the seafood is seriously overcooked.
Why not just season your seafood with the same spices you’re using in the main dish (and add it at the end) and use fish stock for the rice? It accomplishes what you’re after. The only seafood that really absorbs external flavors is scallops, IME, so I don’t think you’re losing much by cooking the seafood separately. I prefer the shrimp to taste like shrimp and the fish to taste like fish, as it offers a counterpoint to the rice seasonings.
I don’t see how you’re going to achieve your goal otherwise. Overcooked seafood is a crime against humanity, and unavoidable if you’re tossing it in too early. What you’re gaining in flavor from the rice seasoning is minimal, IMO.
How is this handled in paella, btw?
Anyways, fish stock it will be (believe it or not, I hadn’t done it). I would much prefer shrimp stock, but whole shrimp is hard to find around here.
I have also found that adding the veggies too early makes them too mushy, so I might make some fish/veggie stock just for this and have fresher firmer veggies in the rice.
I’ve made a lot of paella. Penelope Casas is generally considered to be the cookbook guru for the dish, but I completely disagree with her cooking method, which is to add in the meats and seafood and then put the dish in the oven until the seafood disappears and the chicken turns to rubber. I don’t know what the hell she’s thinking, but if that’s the authentic/traditional preparation method for the dish, the Spanish are missing the boat completely.
I cook the arborio in my liquid of choice, be it fish or chicken stock, along with the onions, shallots, garlic or whatever, and the herbs/spices/saffron. For chicken or shrimp or sausage, I saute or grill it separately until browned, then add it to the dish at the appropriate time so it will be done at the same time as the rice. Shrimp takes surprisingly little time, as do scallops (which I sear ahead of time). This is more difficult than you might think, because you want a crust to develop on the bottom of your paella, but don’t want to burn it.