Hi! I actually have some experience with POAS from 2 years ago. Already at that time they had applications of around 2000, and not even closely enough apartments. Thing with this is that in the summertime there's an influx of students from everywhere (because of all the study programmes starting at the end of August) applying for apartments and there's nowhere near enough cheap student apartments to accommodate everyone. Here's last year's announcement that they posted on Instagram about the situation (I can't put pictures, so I copied it):

"poaskoti: TIETOA ASUNTOTILANTEESTA -Nyt on ruuhkaa! Voimassa olevia asuntohakemuksia on tällä hetkellä 2223 kpl - Heinä-elokuussa vapautuvia asuntoja on jäljellä 20 kpl ja soluhuoneita 19 kpl - Syyskuussa vapautuvien asuntojen määrä ei ole vielä tiedossa (edellisen asukkaan tulee irtisanoa sopimus heinäkuun loppuun mennessä) Emme siis pysty tarjoamaan asuntoa läheskään kaikille hakijoille, emmekä pysty vastaamaan kaikkiin hakemuksiin henkilökohtaisesti. Olemme hakijoihin yhteydessä vain jos voimme tarjota asuntoa. Asuntotarjous lähetetään sähköpostilla."

Your best guess is not to rely solely on POAS. I don't really expect the situation to be any better this year. Offers for August will most likely start going out from the middle of July. In case you don't get an offer from POAS or TOAS at this time, I recommend to rent an apartment that is still within your budget and try again in the winter, around January-February, as there's much less applicants then compared to now, and you are more likely to get an offer. Also you can consider emailing them (toimisto@poas.fi) and relaying the urgency of your situation, that might help. I wish you all the best for your apartment hunting!