I have anaphylaxsis to milk, tree nuts, peanuts, and seafood (fish, shellfish; bascially, animal in water = bad).
A couple tips:
I make allergy cards, too. I use MS Word and make them 4x6" size. I print one (well, I have a couple backups) on normal 4x6 photo paper and then laminate it; that one stays with me and I always carry it in my purse. I will bring it out in a restaurant, but prefer to not let it leave the table, because...
I print a whole bunch onto the plain side of pale yellow 4x6 index cards. I leave the lined side blank so restaurant staff could write on it if they want. I carry a byunch with me and when I travel have even more in my carry-on bag in case I need to replenish my purse supply. These ones the staff can take with them and give to the kitchen and if they don't give them back that is fine (I let them know that). 99% of the time they will take them, about 30% of the time they will bring them back.
I have had (not at Disney) chefs come out of the kitchen to thank me for having the cards as it made their job so much easier. Likewise managers coming to thank me. Most of the time the card seems to be appreciated.
another tip: once you receive your food, ALWAYS VISUALLY CHECK IT. As a PP noted, mistakes CAN happen. I have had mistakes happen at Disney (WDW and
DCL). But I caught them before eating the food because I visually checked the food and saw something was wrong. So, pull apart the hamburger to make sure there is no cheese hiding in it; check that the breakfast potatoe casserole thing doesn't look the same as your sister's whose has dairy in it, and check that salad to make sure there is not cheese grated on it. Obviously for your own allergies visually check accordingly

It won't catch every possible problem, but may catch some of the bigger ones.
That said out of the many many meals I have eaten with Disney, I can count on one hand the number where something has gone wrong.
As for where to eat: I agree you best bets for QS are probably the hotel resorts, for the most part. There are some exceptions though: for example, at the marketplace QS at Beach Club, they have very limited preparation options. I was not able to get a safe hot breakfast there.
We have had some great dining experiences at TS restaurants: Tucker House (breakfast); Boma (breakfast); Whispering Canyon; Spirit of Aloha, Kona, and others. At MK, we have consistently had a good QS experience at Cosmic Ray's: I actually had a great talk with one of the managers there one time as I placed my order (it was a slower time and nobody else waiting for him).
You can contact WDW Special Diets in advance of your trip and work with them. They also have a notification form they may send you; it lets you tell them where your ADRs all are and notify them of the allergies. I always add to it what QS we plan to eat and and when, or for example add a general note that "we are stayaing at Wilderness Lodge and will eat multiple breakfast, lunch, and dinner at Roaring Forks during the course of the trip".
Finally: ALWAYS, EVERY TIME you order food, let the CM know there is a food allergy. This should kick in their allergy protocol. I"ll tell you a little story about why this is so important, because I had a near miss.
We were staying at WL. We arrived mid afternoon and were hungry so went to Roaring Forks to grab some food. I met and had a great talk with the manager. We ended up going over their entire menu and ingredient book, not just for that meal but for future meals so I could preplan. Including breakfast for the next morning. The eggs were listed as just eggs.
Come morning, I am in line to order my breakfast, which I know from reading the ingredient list will be safe if I exclude certain things. But eggs, bacon and potatoes are fine. I wasn't going to say anything about my allergy, just order. At the last second before ordering I changed my mind and said I have a food allergy. Food allergy protocol kicks in and I get to speak with the supervisor. I explain my allergies and what I want. She says no problem, but explains the eggs and potatoes have BUTTER ADDED TO THEM TO MAKE THEM MOISTER (!!!!!!!). Not in the ingredient list ! She made me safe food. If I had not spoken up, just gone off the menu and the ingredient list, at best I would have had a trip to the ER.
As an aside, this actually leads to the concern I have about the "allergy menus" they have introduced. If someone with an actual severe allergy was to rely on those menus alone to place their order, they may end up in a situation like I was nearly in OR have a cross-contamination situation that wasn't prevented because the person preparing the food didn't know they needed to. There needs to be that human:human communication about the allergy and the food. Otherwise, I think, in the long run it is inviting trouble.
We visit WDW and cruise on DCL not just because we enjoy it, but in huge part because they set the gold standard for how food allergies should be handled. I know that with high certainty my food will be safe. I also know there are no guarantees, so I have to do my due diligence as well: inform, ask questions, carry Epipens and asthma rescue inhaler always, visually check the food I receive to make sure it passes the "looks right" "test", and be prepared to walk away if it doesn't feel right (eg chef is clueless).
Traveling with allergies can be a challenge -- I've gotten stuck at Chicago O'Hare for eight hours due to a flight connection cancelation and been refused food service everywhere I tried but McDonalds (which I now can't eat at either); but the McDonalds manager was GREAT, especially considering I was crying by the time I made to his counter after six or eight refusals one after another in an hour -- but also quite rewarding -- safe restaurant pizza at Pizzafari after not being able to have any for years !
One final trip: whether traveling by air or driving, bring some safe non-perishable food (if crossing borders, commercially pre-packaged and still sealed is a really good idea; also be aware of restrictive rules re: fruits, veggies, meats, etc). I like Enjoy Life products: free of top 11 allergens, portable, etc. I have had many times where I have had to delve into my safe snacks for some reason or another, and been very glad I had them, whether during travel itself or at my destination.
SW