How do I deal with postdoc acceptances that come in before my preferred institution makes its decision known?
This can be difficult. If you get an offer with an uncomfortable deadline, I would recommend bringing it to the attention of departments which you prefer (and/or individual faculty at those departments).
This is not guaranteed to push them to make a faster decision, but it could.
It is perfectly ethical to apply to several positions at the same time. It is a different thing if you accept and then decline later on (as they may plan for you to come at that stage); still it happens, but it's not ideal.
The fairest solution is, once you get an acceptance from the first batch, to ask them to wait for a certain period - they may not like it, but you agreeing to getting interviewed is not a promise to join them and they have the option to deadline you if they need a quick decision. If you are strong, they may be more flexible in that respect.
You will not get entirely around this dilemma of balancing your and their legitimate interests.
This is an typical situation in application process with no perfect solution available. I is totally ok to apply for several positions at a time. The institutions are also talking to several applicants in parallel. But after signing a contract, it is not a nice gesture to revoke this w/o reason (even if notice periods would allow this legally). The institute has set trust in you and canceled the application of others. How would you feel if, after you have signed at an institute and canceled other applications, they would say : "Oh sorry, we got the confirmation from somebody which is more suitable to our business targets. Thanks for your time and good luck."
Below are some ideas on how to handle the dilemma. It is up to you, which one you want to follow:
Accept that in life you have to make decisions at time when you don't have all pros and cons available. It's is not like finding a solution to an exam, where the data available clearly defines which answer is correct and which is wrong.
Accept that, even if if you would have all offers available at the same time, you cannot say which offer is the best. Many unexpected events might happen at one position or the other: Your project might get canceled during your work. Your professor might change department. You might meet your future spouse.
If you are really sure which is the best position for you, but the confirmation from this position has still not arrived: Contact this potential employer and explain your situation. You are really interested in this position, but you have also agreements from other institutes so you need to make about the alternatives an decision. Is it possible for your target employer to give you an confirmation earlier? If the employer is really interested in you, they will try to shorten the decision time. Otherwise there is the risk that they lose you.
After signing a contract, withdraw all applications to other institutes. So you save them work. Additionally you avoid you dilemma, when you get an positive reply, but have signed already elsewhere.
Attention: Only be sure to have a job after you have signed a contract. Don't rely on verbal promises only. Things might change easily: The project start might be delayed by 1 year. Or the nephew of the CEO might be interested in the position ...
Asking for time (e.g. 1 weekend) to think it over an recent contract is ok. But at durations longer than 1 week, the employer might become impatient.
A bit more time can be gained, if you ask for some small modifications on an proposed contract. The employer might need some time (eg. 1 week) to issue a new contract. Anyhow this point (and the previous one) shall not be exploited too much. The employer might get the impression that you are not really interested and hire an other candidate in the meantime.
If possible, you can try to reduce the risk a bit, by smart timing of the applications. First send them out to one or two institutions which are not in your main target area. The benefit here, is that you have the opportunity to get some more training in the application process. You become more confident in interview situation. Second try your number 1 choice. You can hope for earlier feedback if there is an opportunity for you. With some delay you could contact departments which are a good fallback solution, althrough not the main target. The idea here is that you are not forced to early to decide to take or leave an offer here. If your number one declines, you might get some positive feedback from these good fallback solutions later. Of course this strategy has to be adapted to time restrictions, and the application deadlines for the adverticed positions.
Career is not only defined by your first job. Of course this first job already gives you an direction in industry you will develop. Further corrections here might be energy and time consuming. But the final result, if you become an owner of an multinational enterprise, an university professor, or simply an employe w/o significant career step depends mostly on personality.
Potential employers prefer candidates who appear to be authentic. Don't try to play to much games with them.
Trust your gut feeling.