I went through this in the past 8 months. In my situation my house was already paid off, so the paying two mortgages wasn't an issue, but there's still risk associated with not selling. Not to mention you have to take care of two houses in the mean time.
The biggest problem I ran into was with the city. In Michigan we have a homestead tax exemption for a primary residence. When I revoked the exemption on the old house and moved it to the new house, the city immediately sent a notice that since my old house was no longer owner-occupied, I had to register it and have it inspected and brought up to the city's rental housing code. I did NOT want to do this, being a 100 year old house it was likely they would cite some violations and then I would be on the hook for fixing them or face condemnation. Long story, but the drama went on for a while and I got the city council involved. My point is, one thing to consider is what are the rules are for vacant housing in your municipality. And don't trust asking a real estate agent, talk to the municipality directly.
I ended up putting my house up for sale in January and the sale closed in late April, so the three month time frame is definitely possible, especially since the January-February timeframe is typically not a good time to sell a house. It really depends on how the real estate market moves in your area, though. My wife's house took about 2 years to sell. I also have a coworker who owned two houses for two years because one didn't sell, but that was during the middle of the housing crisis. Trying to maintain two homes over the summer would be a bear, with everything else you have going on, trying to move and then having to mow two lawns, etc.
The other options aren't always the greatest either. You can try to buy with a contingency based on selling your exisiting place, but the sellers of your new home might not accept that contingency. Or if the buyer's of your house fall through (e.g. poor financing) then you're in trouble. The people I bought my house from went this route and they were very concerned about my creditworthiness because they didn't want the sale to fall through or they'd lose the house they were buying. Then they had a very tight schedule for moving since they closed on both houses the same day. Their move did not go smoothly.
Another option is to sell your house and rent for the interim. That wasn't a good option for us with two dogs and two cats, plus a ton of junk we'd have to move twice.
It wasn't stress free, but in the end it worked out for use to buy before selling, but the main question is the risk of what happens if you can't sell in the time frame you expect.
