I have done Cox regression but there's a suggestion to look into log normal regression.

In the context of analyzing times to events, that suggestion probably meant using a parametric log-normal survival model. That’s implemented directly in standard survival software, for example the R survival package via its survreg() function when called with the argument dist = “lognormal”.

In your situation, results would be conditional on someone having presented for diagnosis. They wouldn’t apply to people who had the same symptoms and disease but didn’t seek medical attention. That could make a big difference for interpretation.

A log-normal survival model is based on specific assumptions about the distribution of baseline survival times and the associations between predictors and survival times. It can be better to use a more agnostic approach via ordinal regression, as explained by Frank Harrell in Chapter 25 of Regression Modeling Strategies. That approach encompasses many specific types of survival analysis and allows you to evaluate which one best describes your data.