What is probably happening here is you have power to the light on one black, another black to the switch, and a return path along the whites. That is, the switch is AFTER the light.
This is generally unsafe as the light is still energized, even when the switch is off, and should be fixed regardless.
The first thing I'd do is beg/borrow/steel a non-contact voltage detector.
With everything still attached, switch the light off. If I'm right, the detector would read hot on BOTH black wires, and the whites will be un-energized. Flip the switch and the whites should be energized as well.
If this is indeed the situation, the fix is relatively easy. Turn the power off! Remove the existing light.
Figure out which pair is wired to the switch. The easiest way is to use a meter in resistance mode. Find the black-white pair that shows low resistance when the switch is on, and open circuit when switched off. Mark the white wire with a loop of electrical tape. This will be the switched hot.
Now wire the blacks together with an appropriate wire nut, and make sure it's a secure connection. You should not be able to see any copper.
Wire the switched hot (the white wire marked with tape) to the black wire of the new fixture. Wire the remaining white to the white wire of the fixture. Make sure you've grounded the fixture according to the manufacturers instructions.
Mount the fixture, and turn the power back on.
The first rule of novices replacing fixtures (or anything) is only mess with the wires that go to the old fixture. Don't touch things that are outside project scope, no matter how interesting they seem. That is to say, resist the urge to learn electrical by dismantling your house. Get a DIY book on the subject and read it through.
Safety Ground wires don't need much thinking, because they all go together.
The clump of all-white wires is certainly neutral, and an appropriate place to terminate the white wire(s) from your fixture. What is always true is that the place the old lamp's white wire went, is correct -- barring previous incompetence of course (and novices are in no position to judge competence).
What remains is the switched-hot wire(s) from the fixture, presuming you want the lamp to be switchable. If that wire comes from the ceiling box, it is usually a solitary (connected to nothing else, who else would want a wire that is only energized with the lamp on?) This will be the wire that went to the lamp's black wire before.
Color coding is often not that helpful. Switched-hot is a type of hot, and all hot wires can be any color not reserved for neutral or ground. Installers get no choice of color since all cables come only black/white or black/white/red. However by convention it's preferred to use red for that when possible. So red is a good bet, and since you say the old lamp's wires went there, that all jibes.
Black and white
Your idea of connecting both lamp whites to the white cluster, and one black to supply black and one black to supply red -- the result was one lamp is always-on, and the other is switched. You may have missed that if you weren't paying close attention. (Light works on a log scale like sound, so half the light is only 3db dimmer. 3db isn't a lot).
In this LED age, it costs so little to run small lights that I have started wiring some lamps to be always-on for safety and put small 3-4 watt LEDs there. ($3-4 a year). In a 2-lamp fixture I'd make one lamp switched and put a very strong LED in it, so you have low/high instead of on/off.
If it also has switched receptacles
Lastly, if you have split bulbs black and red, and it still doesn't switch, there's a fair chance the house also has switched receptacles... and a novice (no names mentioned) replaced the receptacles and wasn't paying attention to the breakaway "tabs" between the two sockets on the hot (brass screw) side. That would defeat the switch, causing the red wire to be always-on.
The cure is to search all the receptacles in the room, looking for red wires or anything abnormal. In such a location (there may be several), get some spare receptacles (they are $3) and break the brass-screw-side tab off. If doing this causes downline receptacles to stop working on both sockets, it was the wrong thing to do; tabs can't be de-broken so swap in a spare receptacle. Needles to say, cut power at the breaker and confirm it's off before opening up a receptacle.
Best Answer
Remove the metal cover from the fixture. There's a black ballast inside that is probably causing your problem. Take some pictures of the wiring and then disconnect all the wires going to the ballast, after turning off the power. Bring the ballast to a home store or electric supply store for a replacement. Then just re install it. A bit of friendly advise: just buy a new fixture as it will probably be cheaper than buying the ballast.