Make sure your AVREP is up to date with the times you are able to deploy.
Hi, with experience as NRD TrgO, as FSTO writes, update your AVREP. Do this as early, and as often, as you can or need to. The AVREP is the tool that your Career Manager shop at NAVRESHQ will use to look for people to fill billets. Be specific with what you want in your availability and in the remarks, e.g. NETP and whatever other courses/packages you're trying to complete.
Also, at most NRDs you should be able to approach the TrgO directly. It's the TrgO's job to both support your training, and assist in employment opportunities where available. The TrgO will have, or be able to find, the current deployment/employment opportunities, and will likely have already advertised them to your unit. If your own unit TrgO is away, or doesn't have the answer, ask to be put in touch with another TrgO from your region and they will help as best they can. As I say, it's the TrgO's job to do just the sort of thing you're trying to achieve here.
You also write that you spoke to someone in your Training office already, who said that you can't really pick where the Navy goes. That's true. However, as a reservist, you have to examine your personal situation and goals as they relate to the Navy's expectation that you will become trade qualified. Apply those criteria to where you want to sail (and what you put in your AVREP). If your goal is to get your package done, NETP, and become eligible for promotion, there are PLENTY of short and long-term sailing opportunities, on several classes of ship. You can start weighing off many signatures simply on ORCA NRETS events that your unit or region will organize. If your goal is a medal, however, as you have also indicated, then the opportunities for participation in a named Operation deployment as an untrained/unqualified sailor are perhaps less frequent, and you may need to execerise some patience. The goals of training and participating on named Operations can be achieved simultaneously, however, just be aware that simply sailing whenever/wherever the Navy needs you could be a much faster route to qualification.
In the meantime, make sure you take advantage of all training opportunities at your unit (C7/Sig requals, FORCE Test, First Aid... also lectures or DLN courses like GBA+, OPERATION HONOUR...). This will ensure that when the time comes, you're as close to ready to go and qualified as you can be.
TLDR:
1. Update your AVREP
2. Talk to your TrgO
3. Go to Sea