You can do a very simple split in the Dartington ... you just put a division board in the middle then apportion the frames between the two halves and open the second entrance. Your queen will be in one half (if you can find her all the better) and as long as you have eggs/1 day old larvae on frames either side of the division board then they will build a queen cell or two in the half without the queen cells. It's not going to stop them swarming but as long as you don't already have queen cells being built it may well slow them down a bit. As the brood emerges from the frames on the side without the queen swap them for frames with brood from the other side to give your existing queen space to lay ... bit of a juggling act required ...
Unfortunately, you do really need another hive ... at least a nuc ... once you have a queen cell in the half without the queen you can transfer that and a few frames to the Nuc and put some more frames in the Dartington and remove the division board ..
I found this in the second season with my long hive as they do build up very quickly. I built one Nuc and was given another and these got me out of a hole but ... it rapidly became obvious that I needed another full hive and not having the time to build another long hive I invested in a Paynes 14 x 12 Poly ... and I must admit that, next to the LDH, they are about as ideal a hive as I could find. I now have several.
The alternative is to look out for another second hand Dartington (new ones are over £339 so not the most economic to buy) - they are not that popular these days and you could probably pick up another one for not a lot - they do come up occasionally on ebay and gumtree but you have to just watch out for them.