Two things to start with... a temperature strip on the outside of your bucket is not telling you what is going inside the bucket. Yeast when at their active peak generate heat. The temperature inside your wort may be 10 degrees or more higher than that strip is telling you. The other thing is that by the time you moved that bucket the bulk of fermentation (at those warm temperatures) was done.
That being said however... relax! You are doing the same as countless other homebrewers when they began brewing. Your beer is just fine. It will turn out fine and taste great.
Controlling fermentation is something that will help make your beer turn out even better in the future but there is no need to go spending a lot of money chasing that goal. You said you have a room that stays in the mid 60's so you already have viable solution. It means your fermenting wort will probably be in the mid 70s which is on the upper end of many yeasts.
You can improve your control by using that cool room you describe without spending a lot of money by using what is called a swamp cooler. It's a tub of some sort that you fill with a little cool water. Place your fermenter in the tub and wrap it with a wet towel or T shirt. You can do only this and drop the temperature by a couple of degrees... add a frozen 2 liter bottle of water or two and you can reduce the temperature of your wort by 5 to 10 degrees. It does take a bit of time and effort to check on a couple times of day but it does work. (link to an article below)
Another thing you might want to consider is getting a bung for your fermenter that is also fitted with a thermowell. With a thermometer that has a long enough probe to fit the thermowell you can now monitor the temperature inside the wort.
There are many ways to control fermentation temperature and not all require spending a lot of money on shiny gadgets.
https://homebrewacademy.com/swamp-cooler-homebrew/