I've bought and I've sold rentals in southern Indiana with tenants. I've sold a rental (duplex) in Fort Wayne with tenants on both sides of the duplex. You don't say if you've been inside the rental to look it over. If possible, I'd have the current Owner ask the tenant to lock in their rent by signing a year or 6 month lease and see what happens even though that would lock me into below market rent a while longer.
Here's what I'd do if it was me:
1. Learn as much as possible about the existing tenant from the current Landlord/Owner. I'm guessing the current Owner manages the property not a PM. Learn why there is not a lease and how well the tenant and landlord relationship has been over the years. I'd learn the deposit status. I'd then make a tentative decision about how much I'd like to keep the tenant vs how much I'd like to increase the rent. Keeping a long term tenant might be worth it to me versus cleaning up after a 10 year stay, even if the rent is below market. I'd also contact the sewer provider (unless it's a septic system) and find out if the sewer bills are current. You will be responsible to pay them if the tenant doesn't pay and if the current landlord leaves them unpaid. I'd learn who the utility providers are, whose name the utilities are in, and if they are current IF the utility companies will tell you which they may not.
2. In my opinion, the new Owner news should come from the current landlord first. That's what I did. Regardless, I'd write an upbeat letter and U.S. Mail it, informing I'm the new landlord as of some date and including instructions on how to pay the rent and what the status is on any existing deposit. If it was me, I'd ask them to let me know in the next couple of weeks if there are any problems to be fixed. I'd give them my contact info and let them know I'll be calling after Closing to "say Hello," (and i'd get their # from the current landlord.)
3. I'd talk w/them and get a feel for their circumstances if they talk freely but I would not pry. I'd discuss if I have a spare key for emergency access. Then, after that, I would know how to verbally give them my thoughts on the amount of the rent. Assuming I decided to go ahead and raise the rent at that point, I'd let them know and I'd give them some reasons why. I'd discuss how it would benefit them to lock in the amount with a lease. Getting them on a lease would be more important to me than the actual amount of rent. Depending on how that goes I'd follow up with a copy of the lease for them to sign, sent by US Mail or handed over in person. The US Mail would include emergency contact #(s) and a letter informing of the rent amount and how they've agreed to pay and the due date. I'd may send a copy of the rent increase letter by certified mail but maybe not, depending on what all i'd learned so far.
I'd always be respectful and positive and stress that the house isn't perfect but my duty is to fix what is broken and their job is to pay on time. I'd give them 2-3 months notice of the rent increase and definitely a minimum of 30 days.
In my experience, Indiana is landlord friendly which helps landlords keep the property in good repair. Mainly you're responsible for the sewer bill and to provide heat and hot water, but not things like air conditioning.
If this was winter, I'd be concerned about pipes freezing if the relationship isn't going well, but a 10 year tenant probably won't let that happen.
I don't really know about the downsides of not having a lease. I suspect as long as you give at least 30 days notice you can raise rent or evict for any reason or no reason at all.
My hope is that the tenant is happy to have saved $ all these years being below market and that the tenant will continue with you at a market rate. Giving them the 3 choices you noted above should be helpful and keep you on good terms with them.