Assuming the big station is a drop off station, clearly the biggest thing is that you just need more platforms, that is why your input is clogging. Also balancing your network on the exit of your station is doing it very backwards; your trains should be balanced when they enter the station, not after leaving. Your biggest limit on station throughput (especially with ro-ro) is how many trains you can get into it, not out
The hardest part about a network that allows merging traffic to pick any line is that inevitably most trains will pick one over the other, leading to one line being over saturated and the other mostly empty. If you want a well-flowing network utilizing all available lines (without resorting to crazy coop contraptions) I think you'll always have to do a bit of manual "nudging" to force some trains onto different lines to maintain the balance and overall throughput. One easy trick is to put a backwards facing path signal on one of the entrances, like so:

- Baker & Co., 2062-02-05#1.png (33.18 KiB) Viewed 3495 times
The inner track was getting flooded with too many trains, so by placing this signal here (thereby increasing the pathfinder cost) most merging trains now take the outer track which is more empty up to this junction, therefore allowing better traffic flow on both main lines. Nothing is "forced" on one track or the other, but a little nudge like this in some places can help a lot in high traffic situations to avoid too many trains flooding one line or the other

Alternatively you can do as LaChupacabra and simply force some or all trains onto certain tracks, which is better for overall traffic flow but does require you to manually maintain the number of trains on each line.
Also since you asked about how to use pre-signals, here is an example of an inline priority merger. This is a great use of pre-signals to help increase traffic flow on highly congested networks:

- prio-example.png (58.47 KiB) Viewed 3495 times
What it does is allow trains on the vertical line to keep flowing through as long as one enters the priority block before the previous one leaves it, therefore forcing trains on the other line to wait for a sufficient break in the traffic before continuing. This helps prevent the constant stop-start action of trains from both lines trying to zipper-merge together, which is terribly inefficient (at least here, real life traffic is a bit different

). These things can be made longer so the priority is held longer with a bit more work and a different design, but that's getting a little deeper into the coop stuff so I'll skip it for now
Also shown there in the blue circle is a simple switch to divert traffic from the inner line to the much-less-used outer line in case trains are stacked up due to the merger. This obviously helps keep trains flowing and can be useful in situations like this where far more trains use one line over another, but should be used sparingly since allowing trains to switch tracks can cause more slowdowns than it solves. In this case being able to divert to the outside line when there's a backup on the inner one proves very useful, so can be a valid tactic in helping overall throughput and load balancing on multiple mainlines, but it depends on your situation
As far as switching to a terminus station, it will take up less space but even a well-designed terminus station will incur at least a ~5-10% decrease in throughput, just based on the fundamental fact that trains have to exit over the same tile they enter. Not a huge problem but something to be aware of

As for design ideas I'm quite fond of the basic design that Alberth explained, though I do it just a bit differently:
This is a bit of an extreme example but does show essentially the same principle behind Alberths example and how you can do multiple lines in and out. This one is 4-in 4-out, but it can be expanded or split up however necessary
Also, a few general points:
- Never use two-way block signals, they will only cause problems. Either make them all one-way block signals or use path signals instead.
- If your platforms are full and you have trains waiting to enter, you either need more platforms or less trains.
- As LaChupacabra pointed out, if you care about throughput get rid of the tight corners. It will take more space but traffic will flow much smoother.
- At your station entrance where the tracks branch out, get rid of the signals in between the path signals, they are unnecessary and will cause trains to sometimes pick a suboptimal path