Finishing off linen seams

I would use just an ordinary zigzag?

I think linen is so lovely I would want to do run an fell seam or more likely a French seam. It would give you an heirloom finish.

A French seam might be bulky, I’d fell them.

That was my thinking too Blossoming and why I suggested both/either. It's also worth thinking about how they will look on the finished garment. A French seam will "look" like an ordinary seam and would be fine in some areas where there is no bulk, but a felled seam can add something to the look of the dress which you may or may not want in that place on the garment.

Would there be enough fabric to spare so that you can do a variety trial seams before actually making up the garment.

Do you know anyone who owns an overlocker and could finish the seams for you? maybe look in Gumtree or Facebook for someone in your area who does alterations and they might finish the seams? Lastly, ask your local dealer if you could hire an overlocker?

Imho, fell or french seams will add to much bulk and could be uncomfortable in some places. A zigzag finish would be my last choice as it's not sturdy enough and won't stop fraying in the long term.

Good luck and hope you find a solution.

a french seam shouldn't be bulky if you trim it before sewing the second time

I'd agree with doing trial runs - you will have enough scraps for that. Although I am cautious about fell seams, they can be a nice feature, and I think might work on a shift.
I agree that ordinary zig-zag doesn't really stop fraying.
I would consider a Hong Kong finish using the selvedges from organza, so they don't add any bulk.

You need some Fray Stop. It’s a liquid specifically for fabric edges and also for gluing fabric together. I quote ‘for a permanent effect, Fray Stop can be ironed’.

Just a thought, it might be easier if you apply Fray Stop to the cut fabric pieces before sewing up.

Or iron strips of lightweight interface to the seams?

I suggest using the overlock foot on your sewing machine. You should get a good finish on linen.

My “good” linens have run and fell seams whereas my cheap linens are simply overlocked-edge “butterfly” seams.