If you are trying to make offers that have a good chance of being accepted, I would suggest the following approach:
1) Make an offer with no closing credits for higher than your realtor's value for the home, and agree to make up a specific amount of the potential shortfall between appraisal and sales price. You will of course choose an amount to cover that makes sense for your situation, including being able to still cover your downpayment and closing costs. If the appraisal comes in at sales price, then that is great, if not, then your escrow has a better chance of staying alive as the seller is not eating the whole shortfall.
If you only have enough money to make a clean offer where you pay your own closing costs, then I suggest making an offer in line with your agent's valuation and cross your fingers. Also, there are some 5% down conventional loans our there right now that have a lower mortgage insurance premium than FHA. Making a conventional loan offer with the features listed above will give you the best chance to begin and escrow and complete one successfully in this CRAZY market.
Thanks and good luck.