Part board leases are available but as mentioned, typically set up by trainers. I’d try couple of barns that offer lessons at the level you are interested in. (Lets face it, there are many levels of lesson outfits.) If their website or FB page mentions that they attend Local A and some A or AA shows, especially if they have riders listed with wins/ribbons in good classes, it is probably not strictly a lesson mill. Go and sign up for a lesson. Tell the trainer what you are looking to do. S/He will want to assess your ability so be prepared to take a few lessons. Be sure to always put the horse and tack away carefully at the level of the barn. (Washing everything and Fig 8’ig the bridle when no others are done might make you look like a snob!) Get to know your lesson mates, especially if they are on their own horses. You might pick up a catch ride that leads to more.
I’ve found that if I’m in a new barn, half leasing a lesson horse is often a way to become friendly with boarders as you’ll probably hack with them etc. If the barn has any events (trainers birthday party, holiday party, potluck dinner etc) make sure you go. Offer to help at shows. Anything you can do to show your interest and competence will help.
ETA: OP, if I were looking to lease out my horse, I probably would skip your ad. You might mention that you’ve just arrived from Ontario, saying “new to country” might make people fear a language barrier. Be specific about your skills (ribbons through Level X in Y Company…) I’d much prefer to hand over the keys to my horse after getting to know a rider and their overall competence first hand or with the guidance of a trusted pro who knows my horse and the other rider as well.
Good luck. There are lots of Marylanders here, I’m sure you’ll get some leads