Certainly the majority of priests begin seminary before the age of 44, but it is not extremely uncommon, I think, for priests to be ordained in their late 40s/early 50s. I don’t think you’re too old, and anyway, there are a couple seminaries which I know accept older candidates. I would speak not only with your parish priest, but perhaps other priests and especially your diocese’s vocation director, so that you have multiple people with whom you can discuss your options. And of course converts can become priests! Often, converts tend to take their Catholic faith more seriously than (many of) their “cradle” Catholic counterparts.
That you have a pretty strong desire and feel in your heart it is the right path could certainly be a sign you are being called to give it a try. Conversely, your priest that you spoke with that told you “not to worry” about it could also be right, although I don’t know how long ago that was - i.e., if it was right around the time of your conversion, or what. If it is still persisting, it could be time to bring up the topic with him again, and if he continues to say the same thing - that you don’t need to worry about it, or something along those lines - as I mentioned, I would speak with some other priests to get their take on it, as well as your vocations director. You shouldn’t dismiss something such as a perception of a possible call to priesthood just because one priest discourages it. Now, if many priests were to discourage you from it, then you might want to begin considering whether or not that is God speaking to you through the Church. Although I don’t know, there might be a lot of people who are against older vocations, for whatever reason, so you’ll need to know why it is that these people wouldn’t think you’re a good candidate.
In any case, keep on praying, and do anything you can (within reason) to be able to attend some daily Masses and do some Eucharistic Adoration. And of course you can receive graces every time you go to Confession as well. If you’re not involved in your current parish, then get involved, especially with the Liturgy if you can (lector, server, choir, whatever).
I often remember priests, seminarians, and those discerning vocations to priesthood and religious life in my prayers…but with a bias towards those discerning vocations to priesthood, since I am in the same boat right now. So you specifically will be in my prayers, at least the next couple times I have those intentions.