Yeah, you can use a palm sander. It's just going to take forever.
Something like this belt sander https://www.amazon.com/7510-01-Sandcat-18-Inch-Pressure-Control/dp/B007A9TD3E/ is probably a good compromise between using a palm sander and using a big floor sander.
The amount of material you take off depends largely on the grit. 30 grit sand paper will take off a lot more floor than 100 git.
Though if your floor is so thin that you're worried about using a floor sander on it, you should probably start saving up to replace it.
A belt sander will eat this off in 2 mins, random orbital sander isn't aggressive enough it's a softy softy approach typically used for finishing. You want one of these https://www.amazon.com/7510-01-Sandcat-18-Inch-Pressure-Control/dp/B007A9TD3E/ but be careful and test it on some scrap first to figure out how quick it'll go. After hitting it with that and taking it back to bare wood, then hit it with the random orbital from 80 grit through to 120 grit step by step ie Belt sander at 40 or 60 grit Random orbital at 80 grit until all marks have been sanded out from the 60 grit Random orbital at 120 grit until all mark have been sanded out from 80 grit And so on jumping up in 20 or 40 increments until it has the finish you want thhhen paint with polyurethane or similar and you are done. Chris Schwartz has a great saying about tools falling into three categories and I'm paraphrasing here :) : Tool that take off a lot of wood ( saws axes etc) Tools that take of a medium level of wood (rasps, belt sanders jack planes etc etc) Finishing tools take off small quantities of wood(files, smoothing planes random orbit sanders etc) :)
What about this SKIL one for $39 I just want an electric belt sander where I can sand knives razor sharp so I'm guessing I need a low grit for changing the grind but a 1000 grit for quick sharpening before honing - https://www.amazon.com/7510-01-Sandcat-18-Inch-Pressure-Control/dp/B007A9TD3E/ref=sr_1_6?s=power-hand-tools&ie=UTF8&qid=1525733941&sr=1-6&keywords=belt+sander
Guess I could buy the $20 WEN rotary tool and try with the highest grit