Seller’s agent’s job is to sell the home at the highest price. OF COURSE s/he/it will say what they think you’ll need to hear to up your offer, even if the “backups” that you’re competing against are illusory.
As far as the market rising rapidly, it hasn’t been since late 2013. Maybe rising moderately. But I doubt that prices will be much higher by summer’s end. If anything, things slow down by August/September.
Lastly, landscaping is irrelevant. You can do it yourself incrementally as time goes on. Look for houses with crappy landscaping, dirty interiors and exteriors. Keeps the schmucks with no imagination out of the competition.
And think with your brain, not your hormones. Schools won’t be a consideration for another few years. No harm in staying pat or renting till you find the right place.