It's better to do the standard $10$-fold cross-validation in your case, since you have $n=260$ observations. Leave-one-out cross-validation is more appropriate if the $n$ is smaller.
You're completely right, after the cross-validation you need to train a new model on the full data with the hyperparameter $\lambda$ found by the cross-validation.
If you're using R for this analysis (which is highly recommended), everything should be easy to implement with the cv.glmnet() function from the glmnet package, with parameter family="binomial" for the logistic regression.
Take care that you normalized your gene expression values reasonably. Look up "RPKM" and "TPM" for a further understanding of how to model with gene expression values.
Below a minimum working example based on your comments, this should give you a smooth start with the cv.glmnet function (X is your gene expression matrix, y your 0-1-encoded disease status):
my_glmnet <- cv.glmnet(X, y, family="binomial") # default CV with 10 folds
plot(my_glmnet) # here you can see the CVMSE and the optimal lambdas, as well as the number of non-zero coefficients at the top
which_opti_lambda <- which(my_glmnet$lambda==my_glmnet$lambda.min) # this gives you the index of the optimal lambda (you can also try lambda.1se)
opti_coefficients <- my_glmnet$glmnet.fit$beta[, which_opti_lambda] # get the coefficients of the final run for the optimal lambda
which(opti_coefficients!=0) # these are the genes that were chosen by lasso cv logistic regression