Welcome to the Dark Side CoCo. Depends a bit on what recumbent you have - some are twitchier than others. I started out on a 20/26" rear-sus Performer, with a relatively low seat height and a seat that's not too reclined. I also have a 700c (no suspension) Performer, and that's much twitchier as the seat height is higher - but quite a bit faster.
I tend to use the 20/26 in the winter when the roads in my part of rural Cheshire/Shropshire are really filthy and bumpy as it's less fidgety than the 700c.
It's really a matter of practice, practice, practice, and choosing your routes. You'd be well advised to get some elbow protectors for the inevitable. Gravel in the elbow is not funny - DAMHIKT. I have an expensive pair as used by the MTB community, and a cheapo pair used by boxers off fleabay - these are not so hot in the summer. Try to avoid the really busy roads, but there's also something to be said for avoiding really narrow roads, as meeting other vehicles on these will inevitably encourage you into the verge to try to let them past, and then you'll be into the rough stuff, with the increased potential for a fall. I had just such a tumble, at 1mph, going uphill and meeting a horse box going downhill. I helpfully pulled well over, only to realise that I needed to stop, and put my left foot down only to find that there was a deep hole in the verge, and over I went into the mud and nettles. The HB drove off..... No harm done, other than dented pride, but it could have been avoided if I'd been more assertive in taking my lane.
A problem I find on a 'bent, compared to a DF, is that your field of view extends further out in front of you, so potholes and gravel right in front (and holes in the verge) tend not to be so visible and I have to be more careful of what's immediately in front of me.
Really, there's no substitute for miles, and some aforethought about routes and what will happen when you do fall off. Long sleeves, longs (or trousers) may come into the equation. It will get easier, but planning your routes may help you to gain experience quickly with less potential for adverse events.
Kim will no doubt be along soon - she has a lot of recumbent experience.