Depending on the season, you could try any of the following, and many more not mentioned:
Pears - try Nashi pears for a change in flavour
Grapes - white, red, black
pawpaw (papaya) - strong flavour, I don't like it myself
dragon fruit (nicer than it looks)
cherries (I prefer the black ones, but the reds are good, too)
oranges (navels are the sweetest)
mandarines (tangerines, clementines)
Rockmelon too strong a flavour for you? Try and find a pepino instead. Similar flavour, but milder, and they come in sizes suitable for one.
If you're getting reactions from apples, don't use them, or apple juice or cider.
Many of the fruits on the list can be cooked and used with things like meats. eg. apricot and chicken, ham and pineapple, rhubarb and pork, grapes in salads, mango with anything. Try frying your bananas and serving with a Chicken Maryland.
You don't have to eat the fruit as a snack. Make fruit salads for desserts, use fruits in steamed puddings, serve fruits in cobblers, or with custard or ice cream or cream or yoghurt, make fruit sauces, or chutneys or jellies. Drink fruit juices, alone or in combinations. Make smoothies. Make pies or cake fillings, or use fresh fruit as a garnish or cake decoration. Make fruit fritters:
Batter For Fritters:
1 heaped tablespoon flour
1 egg, separated
1 tablespoon butter, melted
2 tablespoons lukewarm water
Sift flour into a basin, make a well in the centre and drop the egg yolk into it. Add salt, then gradually add water, stirring the flour in from the centre. Mix until smooth, add the butter. Allow mixture to stand for one hour before using. When ready to use, beat the egg white until stiff, and mix it in lightly to the batter mixture. Add sweet or savoury fillings of your choice: eg. banana, pineapple, apple slices; finely chopped onions and leftover meat, vegetables etc. Drop by tablespoonfuls onto a hot griddle or heavy based frying pan, lightly oiled, and fry other side.
Use fresh fruit with a sweet dip. Make a fondue sauce and dip fruit into it.
Don't forget dried fruit. Sprinkle on breakfast cereals, make muesli, add to biscuit (cookie) or muffin or cake batters. Raisins and sultanas, prunes, apricots, bananas, pawpaw, mango, strawberries, apples, cranberries - all these and more can be bought dried and make good snacks.
Try dates. They can be very sweet, and served fresh, dried, sweet or savoury. Ditto dried figs.
Do try to reduce your intake of sugar. You'll soon get used to it, and then fruit will taste sweeter to you. It's all a matter of training your taste-buds.