Yes, just replace the wires. Even if you reverse the connections, the light will work. For safety's sake, make every effort to connect the wires properly. The silver screw should be connected to the neutral house wiring, white in US and Canada and blue or black nearly everywhere else. Try to confirm the silver screw connects to the outer metal sheath of the socket, the part with the threads. The yellow colored screw connects to the switched power of the house wiring and the center contact in the socket. The switched power house wire color is usually black or red in US and Canada and brown or some other color everywhere else. Just do not attempt to connect any wire with green (may be striped with yellow) or no insulation to either of these screws. Green or bare is for equipment grounding purposes and should be bonded to any metal components that do not carry current.
Strip the insulation from the wire ends no longer than necessary. Try to arrange the wire under the screw head to wrap almost full circle with no overlap. Wrap it in the direction of tightening the screw so the wrap is drawn tighter when tightening the screw. This only applies to binding post screws. If there is a separate hole for the wire and the screw acts like a set screw, then disregard this part. Replicate the connection methods used previously in the junction box, typically either wire nuts or set screw strips.
But please - whatever you do, turn the power off first!
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
If you pulled the wire from the socket then the best way to repair this is to replace the socket unless it happens to have screw terminals on it (I doubt it). If you mistakenly removed a wire nut then you can easily correct this. Wire all black wires together, all whites together, and the grounds together with wire nuts (often they are supplied with the fixture).
The lights should be wired in parallel (white-to-white, etc.), not series. Do not wire black-to-white and black-to-white.
If you are unsure of how to make this repair then the safest thing to do is replace the fixture and/or consult an electrician.