I am a solicitor and I will answer your question today. Please rate my answer positively even if it is not the answer you hoped for, so that I will be paid for my time here.
Unfortunately yes, your convictions will affect your application.
You must disclose both convictions (and any other offences, even if they did not result in a conviction, for example any road traffic offences) in the application form. If you do not disclose an offence or conviction it will appear on the Garda record search that the Department of Justice will conduct and your application will definitely be refused.
If the convictions (and any offences) are properly disclosed, you will be offered an opportunity to explain them. You can write back to the Department to explain the circumstances. If there were mitigating factors, you could ask the solicitor who represented you to provide a letter to explain them. If any of those circumstances were related to a medical condition, you would have had a letter from your doctor for the court at the time - you could provide a copy of that as well to the Department.
Even so, because you were convicted in a court in circumstances where the judge would have had access to all of that information, the relevant fact really is that you were convicted. The Department of Justice (INIS) will, in my experience, always refuse an application here a person has a conviction. The Minister for Justice has absolute discretion under the Irish Naturalisation and Citizenship Act to grant or refuse applications for naturalisation. There is no appeal process. Instead, you will be told that you can apply again at any time. I have corresponded with the Department at length about this, to ask how long a person should wait before applying again, and was told earlier this year on respect of an applicant who was in a similar position to yourself that he should apply again when the latest conviction was ten years old. That is not the law - just an example of the current Minister's approach to your question.
I trust that this has answered your question - although I know it will be a disappointing answer - and that you will rate it accordingly.
Claire McCarthy, Solicitor