Will I be able to get the tourist visa even though I won't be able to show that I'm leaving the country within 60 days?
You really are confused!
A 60-day tourist visa (TV) is something you apply and pay for BEFORE you arrive in Thailand. It's valid for 60 days once activated, and can be extended for another 30 days for Baht 1,900. You cannot get this visa after you have arrived inThailand, it's not available at the airport, nor at any in-country Immigration office.
won't be able to show that I'm leaving the country within 60 days?
That TV has nothing to do with this. You seem to be talking about a visa-exempt entry stamp. This is NOT a visa, it's an entry stamp, if arriving by air, to allow you stay in Thailand for 30 days, but you are supposed to be able to show proof of onward travel from Thailand before that 30 days period expires. It's never a problem arriving in Bangkok without that proof but it may well be the cause of denied boarding of the flight TO Thailand from USA or Canada. Having a pre-arranged 60 day TV gets you around this requirement.
If you CAN board without TV, then the entry stamp will work for you as you say you will leave Thailand again before the 30 days are up. You get another 30 day stamp every time you arrive by air, but it's only 15 days if you enter overland.
Once you leave after your first visit, get a TV in Cambodia or Vietnam if you plan to stay again in Thailand for longer than 30 days.
If you just plan to enter Thailand twice then by far your easiest solution is just get a double entry TV from a Thai consulate in US or Canada before you leave.
You can also keep heading for the border and leave Thailand, then turn around and come right back in again, as often as you like, but again - if you do this at a land border you'll only get 15 days each time, and it costs visa fees for the country you exit into before doing the u-turn.